Lie nilpotent group Lie nilpotent rings algebras and central series Ernesto Spinelli ◮ The lower Lie power series of R is the series Lie nilpotent group algebras and R [ 1 ] ≥ R [ 2 ] ≥ R [ 3 ] ≥ · · · central series Lie nilpotency index Computation of cl ( U ( KG )) whose n -th term R [ n ] is the associative ideal Upper Lie codimension subgroups Open questions generated by all the Lie commutators [ x 1 , . . . , x n ] , with the assumption that R [ 1 ] := R . ◮ The upper Lie power series of R is the series R ( 1 ) ≥ R ( 2 ) ≥ R ( 3 ) ≥ · · · whose n -therm R ( n ) is defined by induction as the associative ideal generated by [ R ( n − 1 ) , R ] , with the assumption that R ( 1 ) := R . ◮ R is called Lie nilpotent ( strongly Lie nilpotent ) if there exists m such that R [ m ] = 0 ( R ( m ) = 0) .
Lie nilpotent group Lie nilpotent rings algebras and central series Ernesto Spinelli ◮ The lower Lie power series of R is the series Lie nilpotent group algebras and R [ 1 ] ≥ R [ 2 ] ≥ R [ 3 ] ≥ · · · central series Lie nilpotency index Computation of cl ( U ( KG )) whose n -th term R [ n ] is the associative ideal Upper Lie codimension subgroups Open questions generated by all the Lie commutators [ x 1 , . . . , x n ] , with the assumption that R [ 1 ] := R . ◮ The upper Lie power series of R is the series R ( 1 ) ≥ R ( 2 ) ≥ R ( 3 ) ≥ · · · whose n -therm R ( n ) is defined by induction as the associative ideal generated by [ R ( n − 1 ) , R ] , with the assumption that R ( 1 ) := R . ◮ R is called Lie nilpotent ( strongly Lie nilpotent ) if there exists m such that R [ m ] = 0 ( R ( m ) = 0) .
Lie nilpotent group Lie nilpotent rings algebras and central series Ernesto Spinelli ◮ The lower Lie power series of R is the series Lie nilpotent group algebras and R [ 1 ] ≥ R [ 2 ] ≥ R [ 3 ] ≥ · · · central series Lie nilpotency index Computation of cl ( U ( KG )) whose n -th term R [ n ] is the associative ideal Upper Lie codimension subgroups Open questions generated by all the Lie commutators [ x 1 , . . . , x n ] , with the assumption that R [ 1 ] := R . ◮ The upper Lie power series of R is the series R ( 1 ) ≥ R ( 2 ) ≥ R ( 3 ) ≥ · · · whose n -therm R ( n ) is defined by induction as the associative ideal generated by [ R ( n − 1 ) , R ] , with the assumption that R ( 1 ) := R . ◮ R is called Lie nilpotent ( strongly Lie nilpotent ) if there exists m such that R [ m ] = 0 ( R ( m ) = 0) .
Lie nilpotent group Lie nilpotent rings algebras and central series Ernesto Spinelli ◮ The lower Lie power series of R is the series Lie nilpotent group algebras and R [ 1 ] ≥ R [ 2 ] ≥ R [ 3 ] ≥ · · · central series Lie nilpotency index Computation of cl ( U ( KG )) whose n -th term R [ n ] is the associative ideal Upper Lie codimension subgroups Open questions generated by all the Lie commutators [ x 1 , . . . , x n ] , with the assumption that R [ 1 ] := R . ◮ The upper Lie power series of R is the series R ( 1 ) ≥ R ( 2 ) ≥ R ( 3 ) ≥ · · · whose n -therm R ( n ) is defined by induction as the associative ideal generated by [ R ( n − 1 ) , R ] , with the assumption that R ( 1 ) := R . ◮ R is called Lie nilpotent ( strongly Lie nilpotent ) if there exists m such that R [ m ] = 0 ( R ( m ) = 0) .
Lie nilpotent group Lie nilpotent rings algebras and central series Ernesto Spinelli ◮ The lower Lie power series of R is the series Lie nilpotent group algebras and R [ 1 ] ≥ R [ 2 ] ≥ R [ 3 ] ≥ · · · central series Lie nilpotency index Computation of cl ( U ( KG )) whose n -th term R [ n ] is the associative ideal Upper Lie codimension subgroups Open questions generated by all the Lie commutators [ x 1 , . . . , x n ] , with the assumption that R [ 1 ] := R . ◮ The upper Lie power series of R is the series R ( 1 ) ≥ R ( 2 ) ≥ R ( 3 ) ≥ · · · whose n -therm R ( n ) is defined by induction as the associative ideal generated by [ R ( n − 1 ) , R ] , with the assumption that R ( 1 ) := R . ◮ R is called Lie nilpotent ( strongly Lie nilpotent ) if there exists m such that R [ m ] = 0 ( R ( m ) = 0) .
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) If R is Lie nilpotent, (strongly Lie nilpotent) the smallest Upper Lie codimension subgroups integer m for which R [ m ] = 0 ( R ( m ) = 0) is called the Lie Open questions nilpotency index ( upper Lie nilpotency index ) of R and it is denoted by t L ( R ) ( t L ( R ) ) .
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) If R is Lie nilpotent, (strongly Lie nilpotent) the smallest Upper Lie codimension subgroups integer m for which R [ m ] = 0 ( R ( m ) = 0) is called the Lie Open questions nilpotency index ( upper Lie nilpotency index ) of R and it is denoted by t L ( R ) ( t L ( R ) ) .
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and ◮ Clearly, R [ n ] ⊆ R ( n ) for all integer n and thus, if R is central series Lie nilpotency index strongly Lie nilpotent, it is Lie nilpotent and Computation of cl ( U ( KG )) Upper Lie codimension t L ( R ) ≤ t L ( R ) . subgroups Open questions ◮ A. Giambruno and S.K. Sehgal (1989) proved that the exterior algebra on a countable infinite-dimensional vector space over a field of characteristic not 2 is Lie nilpotent, but not strongly Lie nilpotent.
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and ◮ Clearly, R [ n ] ⊆ R ( n ) for all integer n and thus, if R is central series Lie nilpotency index strongly Lie nilpotent, it is Lie nilpotent and Computation of cl ( U ( KG )) Upper Lie codimension t L ( R ) ≤ t L ( R ) . subgroups Open questions ◮ A. Giambruno and S.K. Sehgal (1989) proved that the exterior algebra on a countable infinite-dimensional vector space over a field of characteristic not 2 is Lie nilpotent, but not strongly Lie nilpotent.
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and ◮ Clearly, R [ n ] ⊆ R ( n ) for all integer n and thus, if R is central series Lie nilpotency index strongly Lie nilpotent, it is Lie nilpotent and Computation of cl ( U ( KG )) Upper Lie codimension t L ( R ) ≤ t L ( R ) . subgroups Open questions ◮ A. Giambruno and S.K. Sehgal (1989) proved that the exterior algebra on a countable infinite-dimensional vector space over a field of characteristic not 2 is Lie nilpotent, but not strongly Lie nilpotent.
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and ◮ Clearly, R [ n ] ⊆ R ( n ) for all integer n and thus, if R is central series Lie nilpotency index strongly Lie nilpotent, it is Lie nilpotent and Computation of cl ( U ( KG )) Upper Lie codimension t L ( R ) ≤ t L ( R ) . subgroups Open questions ◮ A. Giambruno and S.K. Sehgal (1989) proved that the exterior algebra on a countable infinite-dimensional vector space over a field of characteristic not 2 is Lie nilpotent, but not strongly Lie nilpotent.
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and ◮ Clearly, R [ n ] ⊆ R ( n ) for all integer n and thus, if R is central series Lie nilpotency index strongly Lie nilpotent, it is Lie nilpotent and Computation of cl ( U ( KG )) Upper Lie codimension t L ( R ) ≤ t L ( R ) . subgroups Open questions ◮ A. Giambruno and S.K. Sehgal (1989) proved that the exterior algebra on a countable infinite-dimensional vector space over a field of characteristic not 2 is Lie nilpotent, but not strongly Lie nilpotent.
Lie nilpotent group Lie nilpotent group algebras algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Theorem (Passi-Passman-Sehgal, 1973) Upper Lie codimension subgroups Let KG be a non-commutative group algebra. The Open questions following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) K has positive characteristic p, G is a nilpotent group and its commutator subgroup G ′ is a finite p-group.
Lie nilpotent group Lie nilpotent group algebras algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Theorem (Passi-Passman-Sehgal, 1973) Upper Lie codimension subgroups Let KG be a non-commutative group algebra. The Open questions following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) K has positive characteristic p, G is a nilpotent group and its commutator subgroup G ′ is a finite p-group.
Lie nilpotent group Lie nilpotent group algebras algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Theorem (Passi-Passman-Sehgal, 1973) Upper Lie codimension subgroups Let KG be a non-commutative group algebra. The Open questions following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) K has positive characteristic p, G is a nilpotent group and its commutator subgroup G ′ is a finite p-group.
Lie nilpotent group Lie nilpotent group algebras algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Theorem (Passi-Passman-Sehgal, 1973) Upper Lie codimension subgroups Let KG be a non-commutative group algebra. The Open questions following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) K has positive characteristic p, G is a nilpotent group and its commutator subgroup G ′ is a finite p-group.
Lie nilpotent group Lie nilpotent group algebras algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Theorem (Passi-Passman-Sehgal, 1973) Upper Lie codimension subgroups Let KG be a non-commutative group algebra. The Open questions following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) K has positive characteristic p, G is a nilpotent group and its commutator subgroup G ′ is a finite p-group.
Lie nilpotent group Outline algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Open questions Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Open questions
Lie nilpotent group The nilpotency class of the unit group algebras and central series Ernesto Spinelli Let U ( KG ) be the unit group of a group algebra KG . Lie nilpotent group algebras and central series Theorem (Passi-Passman-Sehgal, 1973 + Khripta, Lie nilpotency index Computation of cl ( U ( KG )) 1972) Upper Lie codimension subgroups Let KG be a non-commutative group algebra over a field Open questions K of positive characteristic p. The following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) U ( KG ) is nilpotent. According to a result by N.D. Gupta and F . Levin (1983) for arbitrary associative unitary rings, if KG is Lie nilpotent cl ( U ( KG )) ≤ t L ( KG ) − 1.
Lie nilpotent group The nilpotency class of the unit group algebras and central series Ernesto Spinelli Let U ( KG ) be the unit group of a group algebra KG . Lie nilpotent group algebras and central series Theorem (Passi-Passman-Sehgal, 1973 + Khripta, Lie nilpotency index Computation of cl ( U ( KG )) 1972) Upper Lie codimension subgroups Let KG be a non-commutative group algebra over a field Open questions K of positive characteristic p. The following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) U ( KG ) is nilpotent. According to a result by N.D. Gupta and F . Levin (1983) for arbitrary associative unitary rings, if KG is Lie nilpotent cl ( U ( KG )) ≤ t L ( KG ) − 1.
Lie nilpotent group The nilpotency class of the unit group algebras and central series Ernesto Spinelli Let U ( KG ) be the unit group of a group algebra KG . Lie nilpotent group algebras and central series Theorem (Passi-Passman-Sehgal, 1973 + Khripta, Lie nilpotency index Computation of cl ( U ( KG )) 1972) Upper Lie codimension subgroups Let KG be a non-commutative group algebra over a field Open questions K of positive characteristic p. The following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) U ( KG ) is nilpotent. According to a result by N.D. Gupta and F . Levin (1983) for arbitrary associative unitary rings, if KG is Lie nilpotent cl ( U ( KG )) ≤ t L ( KG ) − 1.
Lie nilpotent group The nilpotency class of the unit group algebras and central series Ernesto Spinelli Let U ( KG ) be the unit group of a group algebra KG . Lie nilpotent group algebras and central series Theorem (Passi-Passman-Sehgal, 1973 + Khripta, Lie nilpotency index Computation of cl ( U ( KG )) 1972) Upper Lie codimension subgroups Let KG be a non-commutative group algebra over a field Open questions K of positive characteristic p. The following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) U ( KG ) is nilpotent. According to a result by N.D. Gupta and F . Levin (1983) for arbitrary associative unitary rings, if KG is Lie nilpotent cl ( U ( KG )) ≤ t L ( KG ) − 1.
Lie nilpotent group The nilpotency class of the unit group algebras and central series Ernesto Spinelli Let U ( KG ) be the unit group of a group algebra KG . Lie nilpotent group algebras and central series Theorem (Passi-Passman-Sehgal, 1973 + Khripta, Lie nilpotency index Computation of cl ( U ( KG )) 1972) Upper Lie codimension subgroups Let KG be a non-commutative group algebra over a field Open questions K of positive characteristic p. The following statements are equivalent: (i) KG is strongly Lie nilpotent; (ii) KG is Lie nilpotent; (iii) U ( KG ) is nilpotent. According to a result by N.D. Gupta and F . Levin (1983) for arbitrary associative unitary rings, if KG is Lie nilpotent cl ( U ( KG )) ≤ t L ( KG ) − 1.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Computation of cl ( U ( KG )) algebras and central series Ernesto Spinelli ◮ A. Shalev (1989) began a systematical study of the nilpotency class of the unit group of a group algebra Lie nilpotent group of a finite p -group over a field with p elements. algebras and central series ◮ Using the idea by D.B. Coleman and D.S. Passman Lie nilpotency index Computation of cl ( U ( KG )) (1968), the attempts by Shalev were based on seeing Upper Lie codimension subgroups if a wreath product of the type C p ≀ H was involved in Open questions V ( KG ) (in fact, according to an observation by Buckley, in this case t ( H ) = cl ( C p ≀ H ) ≤ cl ( U ( KG )) ). ◮ Shalev conjectured that V (( KG )) always possesses a section isomorphic to the wreath product C p ≀ G ′ . ◮ He proved the result in 1990 when G ′ is cyclic and the characteristic of the ground field is odd and A.B. Konovalov (2001) confirmed the statement in the case in which G is a 2-group of maximal class. ◮ Du’s Theorem (1992) gave a great contribution since it reduced the computation of the nilpotency class cl ( U ( KG )) to that of the Lie nilpotency index t L ( KG ) of the group algebra.
Lie nilpotent group Du’s Theorem algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index ◮ S.A. Jennings (1955) proved that if R is a radical Computation of cl ( U ( KG )) ring, the adjoint group R ◦ is nilpotent if, and only if, R Upper Lie codimension subgroups Open questions is Lie nilpotent. ◮ Jennings conjectured that if R is radical, cl ( R ◦ ) = t L ( R ) − 1. ◮ H. Laue (1984) conjectured that if R is radical, for every non-negative integer n , Z n ( R ) = ζ n ( R ◦ ) . ◮ X. Du (1992) proved Laue conjecture’s.
Lie nilpotent group Du’s Theorem algebras and central series ◮ S.A. Jennings (1955) proved that if R is a radical Ernesto Spinelli ring, the adjoint group R ◦ is nilpotent if, and only if, R Lie nilpotent group is Lie nilpotent. algebras and central series ◮ Jennings conjectured that if R is radical, Lie nilpotency index Computation of cl ( U ( KG )) cl ( R ◦ ) = t L ( R ) − 1. Upper Lie codimension subgroups Open questions ◮ H. Laue (1984) conjectured that if R is radical, for every non-negative integer n , Z n ( R ) = ζ n ( R ◦ ) . ◮ X. Du (1992) proved Laue conjecture’s. Let R be an associative ring. For all a , b ∈ R we set a ◦ b := a + b + ab . It is well known that ( R , ◦ ) is a monoid (with 0 as neutral element). The group R ◦ of all the invertible elements of ( R , ◦ ) is called the adjoint group of R . If R = R ◦ , which means that R coincides with its Jacobson radical, then the ring R is called radical.
Lie nilpotent group Du’s Theorem algebras and central series ◮ S.A. Jennings (1955) proved that if R is a radical Ernesto Spinelli ring, the adjoint group R ◦ is nilpotent if, and only if, R Lie nilpotent group is Lie nilpotent. algebras and central series ◮ Jennings conjectured that if R is radical, Lie nilpotency index Computation of cl ( U ( KG )) cl ( R ◦ ) = t L ( R ) − 1. Upper Lie codimension subgroups Open questions ◮ H. Laue (1984) conjectured that if R is radical, for every non-negative integer n , Z n ( R ) = ζ n ( R ◦ ) . ◮ X. Du (1992) proved Laue conjecture’s. Let R be an associative ring. For all a , b ∈ R we set a ◦ b := a + b + ab . It is well known that ( R , ◦ ) is a monoid (with 0 as neutral element). The group R ◦ of all the invertible elements of ( R , ◦ ) is called the adjoint group of R . If R = R ◦ , which means that R coincides with its Jacobson radical, then the ring R is called radical.
Lie nilpotent group Du’s Theorem algebras and central series Ernesto Spinelli ◮ S.A. Jennings (1955) proved that if R is a radical Lie nilpotent group algebras and ring, the adjoint group R ◦ is nilpotent if, and only if, R central series Lie nilpotency index is Lie nilpotent. Computation of cl ( U ( KG )) Upper Lie codimension subgroups ◮ Jennings conjectured that if R is radical, Open questions cl ( R ◦ ) = t L ( R ) − 1. ◮ H. Laue (1984) conjectured that if R is radical, for every non-negative integer n , Z n ( R ) = ζ n ( R ◦ ) . ◮ X. Du (1992) proved Laue conjecture’s. Z n ( R ) are the terms of the Lie upper central series of R , defined by induction as Z 0 ( R ) := 0 and Z i ( R ) := { x | x ∈ R ∀ y ∈ R [ x , y ] ∈ Z i − 1 ( R ) } .
Lie nilpotent group Du’s Theorem algebras and central series Ernesto Spinelli ◮ S.A. Jennings (1955) proved that if R is a radical Lie nilpotent group algebras and ring, the adjoint group R ◦ is nilpotent if, and only if, R central series Lie nilpotency index is Lie nilpotent. Computation of cl ( U ( KG )) Upper Lie codimension subgroups ◮ Jennings conjectured that if R is radical, Open questions cl ( R ◦ ) = t L ( R ) − 1. ◮ H. Laue (1984) conjectured that if R is radical, for every non-negative integer n , Z n ( R ) = ζ n ( R ◦ ) . ◮ X. Du (1992) proved Laue conjecture’s. Z n ( R ) are the terms of the Lie upper central series of R , defined by induction as Z 0 ( R ) := 0 and Z i ( R ) := { x | x ∈ R ∀ y ∈ R [ x , y ] ∈ Z i − 1 ( R ) } .
Lie nilpotent group Application to the Unit Group algebras and central series Ernesto Spinelli Applying Du’s Theorem to group algebras we obtain that Lie nilpotent group if K is a field of positive characteristic p and G is a finite algebras and central series p -group, then Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension cl ( U ( KG )) = t L ( KG ) − 1 . subgroups Open questions ◮ The computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ A.K. Bhandari and I.B.S. Passi (1992) proved that t L ( KG ) = t L ( KG ) under the assumption that p ≥ 5. ◮ Under this assumption the computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ Jennings’s Theory provides a rather satisfactory method for the computation of t L ( KG ) .
Lie nilpotent group Application to the Unit Group algebras and central series Ernesto Spinelli Applying Du’s Theorem to group algebras we obtain that Lie nilpotent group if K is a field of positive characteristic p and G is a finite algebras and central series p -group, then Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension cl ( U ( KG )) = t L ( KG ) − 1 . subgroups Open questions ◮ The computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ A.K. Bhandari and I.B.S. Passi (1992) proved that t L ( KG ) = t L ( KG ) under the assumption that p ≥ 5. ◮ Under this assumption the computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ Jennings’s Theory provides a rather satisfactory method for the computation of t L ( KG ) .
Lie nilpotent group Application to the Unit Group algebras and central series Ernesto Spinelli Applying Du’s Theorem to group algebras we obtain that Lie nilpotent group if K is a field of positive characteristic p and G is a finite algebras and central series p -group, then Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension cl ( U ( KG )) = t L ( KG ) − 1 . subgroups Open questions ◮ The computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ A.K. Bhandari and I.B.S. Passi (1992) proved that t L ( KG ) = t L ( KG ) under the assumption that p ≥ 5. ◮ Under this assumption the computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ Jennings’s Theory provides a rather satisfactory method for the computation of t L ( KG ) .
Lie nilpotent group Application to the Unit Group algebras and central series Ernesto Spinelli Applying Du’s Theorem to group algebras we obtain that Lie nilpotent group if K is a field of positive characteristic p and G is a finite algebras and central series p -group, then Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension cl ( U ( KG )) = t L ( KG ) − 1 . subgroups Open questions ◮ The computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ A.K. Bhandari and I.B.S. Passi (1992) proved that t L ( KG ) = t L ( KG ) under the assumption that p ≥ 5. ◮ Under this assumption the computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ Jennings’s Theory provides a rather satisfactory method for the computation of t L ( KG ) .
Lie nilpotent group Application to the Unit Group algebras and central series Ernesto Spinelli Applying Du’s Theorem to group algebras we obtain that Lie nilpotent group if K is a field of positive characteristic p and G is a finite algebras and central series p -group, then Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension cl ( U ( KG )) = t L ( KG ) − 1 . subgroups Open questions ◮ The computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ A.K. Bhandari and I.B.S. Passi (1992) proved that t L ( KG ) = t L ( KG ) under the assumption that p ≥ 5. ◮ Under this assumption the computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ Jennings’s Theory provides a rather satisfactory method for the computation of t L ( KG ) .
Lie nilpotent group Application to the Unit Group algebras and central series Ernesto Spinelli Applying Du’s Theorem to group algebras we obtain that Lie nilpotent group if K is a field of positive characteristic p and G is a finite algebras and central series p -group, then Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension cl ( U ( KG )) = t L ( KG ) − 1 . subgroups Open questions ◮ The computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ A.K. Bhandari and I.B.S. Passi (1992) proved that t L ( KG ) = t L ( KG ) under the assumption that p ≥ 5. ◮ Under this assumption the computation of cl ( U ( KG )) is reduced to that of t L ( KG ) . ◮ Jennings’s Theory provides a rather satisfactory method for the computation of t L ( KG ) .
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group We set for all positive integer n algebras and central series Lie nilpotency index D ( n ) ( KG ) := G ∩ ( 1 + KG ( n ) ) = G ∩ ( 1 + ω ( G ) ( n ) ) , Computation of cl ( U ( KG )) Upper Lie codimension subgroups Open questions the so called n-th upper Lie dimension subgroup of G . Put p d ( k ) := | D ( k ) ( G ) : D ( k + 1 ) ( G ) | , where k ≥ 1. If KG is Lie nilpotent, t L ( KG ) = 2 + ( p − 1 ) � md ( m + 1 ) . m ≥ 1
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group We set for all positive integer n algebras and central series Lie nilpotency index D ( n ) ( KG ) := G ∩ ( 1 + KG ( n ) ) = G ∩ ( 1 + ω ( G ) ( n ) ) , Computation of cl ( U ( KG )) Upper Lie codimension subgroups Open questions the so called n-th upper Lie dimension subgroup of G . Put p d ( k ) := | D ( k ) ( G ) : D ( k + 1 ) ( G ) | , where k ≥ 1. If KG is Lie nilpotent, t L ( KG ) = 2 + ( p − 1 ) � md ( m + 1 ) . m ≥ 1
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group We set for all positive integer n algebras and central series Lie nilpotency index D ( n ) ( KG ) := G ∩ ( 1 + KG ( n ) ) = G ∩ ( 1 + ω ( G ) ( n ) ) , Computation of cl ( U ( KG )) Upper Lie codimension subgroups Open questions the so called n-th upper Lie dimension subgroup of G . Put p d ( k ) := | D ( k ) ( G ) : D ( k + 1 ) ( G ) | , where k ≥ 1. If KG is Lie nilpotent, t L ( KG ) = 2 + ( p − 1 ) � md ( m + 1 ) . m ≥ 1
Lie nilpotent group Outline algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Open questions Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Open questions
Lie nilpotent group The definition algebras and central series Ernesto Spinelli Let KG be the group algebra of a group G over a field K . Lie nilpotent group algebras and We consider the upper Lie central series of KG , central series Lie nilpotency index Computation of cl ( U ( KG )) 0 =: Z 0 ( KG ) < Z 1 ( KG ) ≤ Z 2 ( KG ) ≤ · · · ≤ Z m ( KG ) ≤ · · · . Upper Lie codimension subgroups Open questions We set ∀ n ∈ N 0 C n ( G ) := G ∩ ( 1 + Z n ( KG )) = G ∩ ( 1 + Z n ( ω ( G ))) . ◮ C n ( G ) is a subgroup of G . We call the i -th term C i ( G ) the i -th upper Lie codimension subgroup of G .
Lie nilpotent group The definition algebras and central series Ernesto Spinelli Let KG be the group algebra of a group G over a field K . Lie nilpotent group algebras and We consider the upper Lie central series of KG , central series Lie nilpotency index Computation of cl ( U ( KG )) 0 =: Z 0 ( KG ) < Z 1 ( KG ) ≤ Z 2 ( KG ) ≤ · · · ≤ Z m ( KG ) ≤ · · · . Upper Lie codimension subgroups Open questions We set ∀ n ∈ N 0 C n ( G ) := G ∩ ( 1 + Z n ( KG )) = G ∩ ( 1 + Z n ( ω ( G ))) . ◮ C n ( G ) is a subgroup of G . We call the i -th term C i ( G ) the i -th upper Lie codimension subgroup of G .
Lie nilpotent group The definition algebras and central series Ernesto Spinelli Let KG be the group algebra of a group G over a field K . Lie nilpotent group algebras and We consider the upper Lie central series of KG , central series Lie nilpotency index Computation of cl ( U ( KG )) 0 =: Z 0 ( KG ) < Z 1 ( KG ) ≤ Z 2 ( KG ) ≤ · · · ≤ Z m ( KG ) ≤ · · · . Upper Lie codimension subgroups Open questions We set ∀ n ∈ N 0 C n ( G ) := G ∩ ( 1 + Z n ( KG )) = G ∩ ( 1 + Z n ( ω ( G ))) . ◮ C n ( G ) is a subgroup of G . We call the i -th term C i ( G ) the i -th upper Lie codimension subgroup of G .
Lie nilpotent group The definition algebras and central series Ernesto Spinelli Let KG be the group algebra of a group G over a field K . Lie nilpotent group algebras and We consider the upper Lie central series of KG , central series Lie nilpotency index Computation of cl ( U ( KG )) 0 =: Z 0 ( KG ) < Z 1 ( KG ) ≤ Z 2 ( KG ) ≤ · · · ≤ Z m ( KG ) ≤ · · · . Upper Lie codimension subgroups Open questions We set ∀ n ∈ N 0 C n ( G ) := G ∩ ( 1 + Z n ( KG )) = G ∩ ( 1 + Z n ( ω ( G ))) . ◮ C n ( G ) is a subgroup of G . We call the i -th term C i ( G ) the i -th upper Lie codimension subgroup of G .
Lie nilpotent group The definition algebras and central series Ernesto Spinelli Let KG be the group algebra of a group G over a field K . Lie nilpotent group algebras and We consider the upper Lie central series of KG , central series Lie nilpotency index Computation of cl ( U ( KG )) 0 =: Z 0 ( KG ) < Z 1 ( KG ) ≤ Z 2 ( KG ) ≤ · · · ≤ Z m ( KG ) ≤ · · · . Upper Lie codimension subgroups Open questions We set ∀ n ∈ N 0 C n ( G ) := G ∩ ( 1 + Z n ( KG )) = G ∩ ( 1 + Z n ( ω ( G ))) . ◮ C n ( G ) is a subgroup of G . We call the i -th term C i ( G ) the i -th upper Lie codimension subgroup of G .
Lie nilpotent group algebras and central series Ernesto Spinelli Lie nilpotent group algebras and Theorem (Catino, S.) central series Lie nilpotency index Let KG be the group algebra of a group G over a field K. Computation of cl ( U ( KG )) Upper Lie codimension subgroups Then Open questions ◮ � 1 � = C 0 ( G ) ≤ C 1 ( G ) = ζ ( G ) ≤ · · · ≤ C m ( G ) ≤ · · · is an ascending central series of G; ◮ if K has positive characteristic p, then, for every positive integer n, C n + 1 ( G ) / C n − p + 2 ( G ) is an elementary abelian p-group.
Lie nilpotent group algebras and central series Ernesto Spinelli Theorem (Catino, S.) Lie nilpotent group algebras and central series Let KG be the group algebra of a group G over a field K. Lie nilpotency index Computation of cl ( U ( KG )) Then Upper Lie codimension subgroups ◮ � 1 � = C 0 ( G ) ≤ C 1 ( G ) = ζ ( G ) ≤ · · · ≤ C m ( G ) ≤ · · · is Open questions an ascending central series of G; ◮ if K has positive characteristic p, then, for every positive integer n, C n + 1 ( G ) / C n − p + 2 ( G ) is an elementary abelian p-group. G = D ( 1 ) ( G ) ≥ D ( 2 ) ( G ) = G ′ ≥ · · · ≥ D ( m ) ( G ) ≥ · · · is a descending central series of G .
Lie nilpotent group algebras and central series Ernesto Spinelli Theorem (Catino, S.) Let KG be the group algebra of a group G over a field K. Lie nilpotent group algebras and Then central series Lie nilpotency index ◮ � 1 � = C 0 ( G ) ≤ C 1 ( G ) = ζ ( G ) ≤ · · · ≤ C m ( G ) ≤ · · · is Computation of cl ( U ( KG )) Upper Lie codimension subgroups an ascending central series of G; Open questions ◮ if K has positive characteristic p, then, for every positive integer n, C n + 1 ( G ) / C n − p + 2 ( G ) is an elementary abelian p-group. G = D ( 1 ) ( G ) ≥ D ( 2 ) ( G ) = G ′ ≥ · · · ≥ D ( m ) ( G ) ≥ · · · is a descending central series of G . If K has positive characteristic p , then, for every positive integer n ≥ 2, D ( n ) ( G ) / D ( n + 1 ) ( G ) is an elementary abelian p -group.
Lie nilpotent group algebras and central series Ernesto Spinelli Theorem (Catino, S.) Let KG be the group algebra of a group G over a field K. Lie nilpotent group algebras and Then central series Lie nilpotency index ◮ � 1 � = C 0 ( G ) ≤ C 1 ( G ) = ζ ( G ) ≤ · · · ≤ C m ( G ) ≤ · · · is Computation of cl ( U ( KG )) Upper Lie codimension subgroups an ascending central series of G; Open questions ◮ if K has positive characteristic p, then, for every positive integer n, C n + 1 ( G ) / C n − p + 2 ( G ) is an elementary abelian p-group. G = D ( 1 ) ( G ) ≥ D ( 2 ) ( G ) = G ′ ≥ · · · ≥ D ( m ) ( G ) ≥ · · · is a descending central series of G . If K has positive characteristic p , then, for every positive integer n ≥ 2, D ( n ) ( G ) / D ( n + 1 ) ( G ) is an elementary abelian p -group.
Lie nilpotent group Du’s Theorem and ULC subgroups algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Let K be a field of positive characteristic p and let G be a Open questions finite p -group. Then ◮ ∀ i ∈ N C i ( G ) = G ∩ ζ i ( V ( KG )) ; ◮ the minimal integer n such that C n ( G ) = G is the nilpotency class of U ( KG ) .
Lie nilpotent group Du’s Theorem and ULC subgroups algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Let K be a field of positive characteristic p and let G be a Open questions finite p -group. Then ◮ ∀ i ∈ N C i ( G ) = G ∩ ζ i ( V ( KG )) ; ◮ the minimal integer n such that C n ( G ) = G is the nilpotency class of U ( KG ) .
Lie nilpotent group Du’s Theorem and ULC subgroups algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Let K be a field of positive characteristic p and let G be a Open questions finite p -group. Then ◮ ∀ i ∈ N C i ( G ) = G ∩ ζ i ( V ( KG )) ; ◮ the minimal integer n such that C n ( G ) = G is the nilpotency class of U ( KG ) .
Lie nilpotent group Du’s Theorem and ULC subgroups algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Computation of cl ( U ( KG )) Upper Lie codimension subgroups Let K be a field of positive characteristic p and let G be a Open questions finite p -group. Then ◮ ∀ i ∈ N C i ( G ) = G ∩ ζ i ( V ( KG )) ; ◮ the minimal integer n such that C n ( G ) = G is the nilpotency class of U ( KG ) .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group A contribution to the conjecture algebras and central series t L ( KG ) = t L ( KG ) Ernesto Spinelli Lie nilpotent group algebras and central series Lie nilpotency index Theorem (Catino, S.) Computation of cl ( U ( KG )) Upper Lie codimension Let K be a field of positive characteristic p. Then subgroups Open questions cl ( U ( KG )) = t L ( KG ) = t L ( KG ) if ◮ G is in CF ( 4 , n , p ) ; ◮ G is in CF ( 5 , n , 2 ) . According to Blackburn’s definition, a finite group G belongs to CF ( m , n , p ) if | G | = p n , cl ( G ) = m − 1 and ∀ i ∈ m − 1 ⌋ \ { 1 } | γ i ( G ) : γ i + 1 ( G ) | = p .
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Remarks and applications algebras and central series According to a result by R.K. Sharma and Vikas Bist Ernesto Spinelli (1992), t L ( KG ) ≤ t L ( KG ) ≤ | G ′ | + 1. Lie nilpotent group ◮ A. Shalev (1993) proved that, if p ≥ 5, | G ′ | = p n for algebras and central series some integer n and t L ( KG ) < | G ′ | + 1, then Lie nilpotency index t L ( KG ) ≤ p n − 1 + 2 p − 1 and the equality holds if, and Computation of cl ( U ( KG )) Upper Lie codimension only if, G ′ has a cyclic subgroup of index p and subgroups Open questions γ 3 ( G ) �≤ G ′ p . ◮ Assume that G is a CF ( 5 , n , 2 ) group and K is a field of even characteristic.Then t L ( KG ) = t L ( KG ) = 8 > 2 3 − 1 + 4 − 1 = 7 . In this sense, Shalev’s inequality does not hold in characteristic 2. ◮ Let f ( 2 , n ) be a function such that t L ( KG ) ≤ f ( 2 , n ) when t L ( KG ) is not maximal. The upper bound is exact when G is a CF ( 5 , n , 2 ) group and G ′ is elementary abelian. In this case G ′ does not contain any cyclic subgroup of index 2, thus also the group-theoretical condition required by Shalev’s result does not hold.
Lie nilpotent group Almost maximal Lie nilpotency index algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series ◮ Shalev (1993) classified non-commutative Lie Lie nilpotency index Computation of cl ( U ( KG )) nilpotent group algebra KG whose Lie nilpotency Upper Lie codimension subgroups index is | G ′ | + 1 under the assumption that Open questions char K ≥ 5. ◮ V. Bovdi and Spinelli (2004) completed the classification in the cases in which char K ≤ 3. ◮ According to results by Shalev and V. Bovdi and Spinelli, if t L ( KG ) is not maximal, the next highest possible value assumed by t L ( KG ) and t L ( KG ) is | G ′ | − p + 2, supposed char K = p .
Lie nilpotent group Almost maximal Lie nilpotency index algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series ◮ Shalev (1993) classified non-commutative Lie Lie nilpotency index Computation of cl ( U ( KG )) nilpotent group algebra KG whose Lie nilpotency Upper Lie codimension subgroups index is | G ′ | + 1 under the assumption that Open questions char K ≥ 5. ◮ V. Bovdi and Spinelli (2004) completed the classification in the cases in which char K ≤ 3. ◮ According to results by Shalev and V. Bovdi and Spinelli, if t L ( KG ) is not maximal, the next highest possible value assumed by t L ( KG ) and t L ( KG ) is | G ′ | − p + 2, supposed char K = p .
Lie nilpotent group Almost maximal Lie nilpotency index algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series ◮ Shalev (1993) classified non-commutative Lie Lie nilpotency index Computation of cl ( U ( KG )) nilpotent group algebra KG whose Lie nilpotency Upper Lie codimension subgroups index is | G ′ | + 1 under the assumption that Open questions char K ≥ 5. ◮ V. Bovdi and Spinelli (2004) completed the classification in the cases in which char K ≤ 3. ◮ According to results by Shalev and V. Bovdi and Spinelli, if t L ( KG ) is not maximal, the next highest possible value assumed by t L ( KG ) and t L ( KG ) is | G ′ | − p + 2, supposed char K = p .
Lie nilpotent group Almost maximal Lie nilpotency index algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series ◮ Shalev (1993) classified non-commutative Lie Lie nilpotency index Computation of cl ( U ( KG )) nilpotent group algebra KG whose Lie nilpotency Upper Lie codimension subgroups index is | G ′ | + 1 under the assumption that Open questions char K ≥ 5. ◮ V. Bovdi and Spinelli (2004) completed the classification in the cases in which char K ≤ 3. ◮ According to results by Shalev and V. Bovdi and Spinelli, if t L ( KG ) is not maximal, the next highest possible value assumed by t L ( KG ) and t L ( KG ) is | G ′ | − p + 2, supposed char K = p .
Lie nilpotent group Theorem AM algebras and central series Ernesto Spinelli Lie nilpotent group algebras and central series Theorem () Lie nilpotency index Computation of cl ( U ( KG )) Let KG be over a field K of positive characteristic p. Upper Lie codimension subgroups Open questions Then the following conditions are equivalent: (b) U ( KG ) has almost maximal nilpotency class; (c) p and G satisfy one of the following conditions: cl ( G ) = 2 and G ′ is non-cyclic of order 4 ; (i) p = 2 , cl ( G ) = 4 and G ′ is abelian non-cyclic of (ii) p = 2 , order 8 ; cl ( G ) = 3 and G ′ is abelian non-cyclic of (iii) p = 3 , order 9 .
Lie nilpotent group Theorem AM algebras and central series Ernesto Spinelli Lie nilpotent group algebras and Theorem (V. Bovdi, S.) central series Lie nilpotency index Computation of cl ( U ( KG )) Let KG be a non-commutative Lie nilpotent group algebra Upper Lie codimension subgroups over a field K of positive characteristic p. Then the Open questions following conditions are equivalent: (b) U ( KG ) has almost maximal nilpotency class; (c) p and G satisfy one of the following conditions: cl ( G ) = 2 and G ′ is non-cyclic of order 4 ; (i) p = 2 , cl ( G ) = 4 and G ′ is abelian non-cyclic of (ii) p = 2 , order 8 ; cl ( G ) = 3 and G ′ is abelian non-cyclic of (iii) p = 3 , order 9 .
Lie nilpotent group Theorem AM algebras and central series Ernesto Spinelli Theorem (V. Bovdi, S.) Lie nilpotent group algebras and central series Let KG be a non-commutative Lie nilpotent group algebra Lie nilpotency index Computation of cl ( U ( KG )) over a field K of positive characteristic p. Then the Upper Lie codimension subgroups following conditions are equivalent: Open questions (a) KG has almost maximal Lie nilpotency index; (b) KG has upper almost maximal Lie nilpotency index; (b) U ( KG ) has almost maximal nilpotency class; (c) p and G satisfy one of the following conditions: cl ( G ) = 2 and G ′ is non-cyclic of order 4 ; (i) p = 2 , cl ( G ) = 4 and G ′ is abelian non-cyclic of (ii) p = 2 , order 8 ; cl ( G ) = 3 and G ′ is abelian non-cyclic of (iii) p = 3 , order 9 .
Lie nilpotent group Theorem AM algebras and central series Ernesto Spinelli Theorem (V. Bovdi, S.) Lie nilpotent group algebras and central series Let KG be a non-commutative Lie nilpotent group algebra Lie nilpotency index Computation of cl ( U ( KG )) over a field K of positive characteristic p. Then the Upper Lie codimension subgroups following conditions are equivalent: Open questions (a) KG has almost maximal Lie nilpotency index; (b) KG has upper almost maximal Lie nilpotency index; (b) U ( KG ) has almost maximal nilpotency class; (c) p and G satisfy one of the following conditions: cl ( G ) = 2 and G ′ is non-cyclic of order 4 ; (i) p = 2 , cl ( G ) = 4 and G ′ is abelian non-cyclic of (ii) p = 2 , order 8 ; cl ( G ) = 3 and G ′ is abelian non-cyclic of (iii) p = 3 , order 9 .
Lie nilpotent group Theorem AM algebras and central series Ernesto Spinelli Theorem (V. Bovdi, S.) Lie nilpotent group algebras and central series Let KG be a non-commutative Lie nilpotent group algebra Lie nilpotency index Computation of cl ( U ( KG )) over a field K of positive characteristic p. Then the Upper Lie codimension subgroups following conditions are equivalent: Open questions (a) KG has almost maximal Lie nilpotency index; (b) KG has upper almost maximal Lie nilpotency index; (b) U ( KG ) has almost maximal nilpotency class; (c) p and G satisfy one of the following conditions: cl ( G ) = 2 and G ′ is non-cyclic of order 4 ; (i) p = 2 , cl ( G ) = 4 and G ′ is abelian non-cyclic of (ii) p = 2 , order 8 ; cl ( G ) = 3 and G ′ is abelian non-cyclic of (iii) p = 3 , order 9 .
Lie nilpotent group Theorem AM algebras and central series Ernesto Spinelli Theorem (V. Bovdi, S.) Lie nilpotent group algebras and central series Let KG be a non-commutative Lie nilpotent group algebra Lie nilpotency index Computation of cl ( U ( KG )) over a field K of positive characteristic p. Then the Upper Lie codimension subgroups following conditions are equivalent: Open questions (a) KG has almost maximal Lie nilpotency index; (b) KG has upper almost maximal Lie nilpotency index; (b) U ( KG ) has almost maximal nilpotency class; (c) p and G satisfy one of the following conditions: cl ( G ) = 2 and G ′ is non-cyclic of order 4 ; (i) p = 2 , cl ( G ) = 4 and G ′ is abelian non-cyclic of (ii) p = 2 , order 8 ; cl ( G ) = 3 and G ′ is abelian non-cyclic of (iii) p = 3 , order 9 .
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