On the Prime Graph Question for 4-primary Groups I Andreas B achle - PowerPoint PPT Presentation

On the Prime Graph Question for 4-primary Groups I Andreas B¨ achle and Leo Margolis Vrije Universiteit Brussel and Universit¨ at Stuttgart Brock International Conference on Groups, Rings and Group Rings July 28 to August 01, 2014

Notations G finite group

Notations G finite group R commutative ring with identity element 1

Notations G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R

Notations G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R � � ε augemtation map of RG , i.e. ε � r g g = � r g . g ∈ G g ∈ G

Notations G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R � � ε augemtation map of RG , i.e. ε � r g g = � r g . g ∈ G g ∈ G U( RG ) group of units of RG

Notations G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R � � ε augemtation map of RG , i.e. ε � r g g = � r g . g ∈ G g ∈ G U( RG ) group of units of RG V( RG ) group of units of RG of augmentation 1 aka normalized units.

Notations G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R � � ε augemtation map of RG , i.e. ε � r g g = � r g . g ∈ G g ∈ G U( RG ) group of units of RG V( RG ) group of units of RG of augmentation 1 aka normalized units. U( RG ) = U( R ) · V( RG )

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g .

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991) � SL(2 , 5) (Dokuchaev, Juriaans, Polcino Milies, 1997)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991) � SL(2 , 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) � groups of order at most 71 (H¨ ofert, 2004)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991) � SL(2 , 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) � groups of order at most 71 (H¨ ofert, 2004) � PSL(2 , 7), PSL(2 , 11), PSL(2 , 13) (Hertweck, 2004)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991) � SL(2 , 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) � groups of order at most 71 (H¨ ofert, 2004) � PSL(2 , 7), PSL(2 , 11), PSL(2 , 13) (Hertweck, 2004) � A 6 ≃ PSL(2 , 9) (Hertweck, 2007)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991) � SL(2 , 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) � groups of order at most 71 (H¨ ofert, 2004) � PSL(2 , 7), PSL(2 , 11), PSL(2 , 13) (Hertweck, 2004) � A 6 ≃ PSL(2 , 9) (Hertweck, 2007) � metacyclic groups (Hertweck, 2008)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991) � SL(2 , 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) � groups of order at most 71 (H¨ ofert, 2004) � PSL(2 , 7), PSL(2 , 11), PSL(2 , 13) (Hertweck, 2004) � A 6 ≃ PSL(2 , 9) (Hertweck, 2007) � metacyclic groups (Hertweck, 2008) � PSL(2 , 8) , PSL(2 , 17) (Gildea; Kimmerle, Konovalov, 2012)

(First) Zassenhaus Conjecture (Zassenhaus, 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman, 1939) � A 5 (Luthar, Passi, 1989) � S 5 (Luthar, Trama, 1991) � nilpotent groups (Weiss, 1991) � SL(2 , 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) � groups of order at most 71 (H¨ ofert, 2004) � PSL(2 , 7), PSL(2 , 11), PSL(2 , 13) (Hertweck, 2004) � A 6 ≃ PSL(2 , 9) (Hertweck, 2007) � metacyclic groups (Hertweck, 2008) � PSL(2 , 8) , PSL(2 , 17) (Gildea; Kimmerle, Konovalov, 2012) � cyclic-by-abelian (Caicedo, Margolis, del R´ ıo, 2013)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle, 2006) (PQ) Γ( G ) = Γ(V( Z G ))?

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle, 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle, 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ) � Frobenius groups (Kimmerle, 2006)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle, 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ) � Frobenius groups (Kimmerle, 2006) � solvable groups (H¨ ofert, Kimmerle, 2006)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle, 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ) � Frobenius groups (Kimmerle, 2006) � solvable groups (H¨ ofert, Kimmerle, 2006) � PSL(2 , p ), p a rational prime (Hertweck, 2007)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle, 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ) � Frobenius groups (Kimmerle, 2006) � solvable groups (H¨ ofert, Kimmerle, 2006) � PSL(2 , p ), p a rational prime (Hertweck, 2007) � half of the sporadic simple groups (Bovdi, Konovalov, et. al. , 2005 – )

Let C ⊆ G be a conjugacy class and u = � u g g ∈ RG . g ∈ G

Let C ⊆ G be a conjugacy class and u = � u g g ∈ RG . Then g ∈ G � ε C ( u ) = u g g ∈ C is called the partial augmentation of u at the conjugacy class C .

Let C ⊆ G be a conjugacy class and u = � u g g ∈ RG . Then g ∈ G � ε C ( u ) = u g g ∈ C is called the partial augmentation of u at the conjugacy class C . Theorem (Berman, 1955; Higman, 1939) Let u ∈ Z G a normalized torsion unit, u � = 1 . Then ε 1 ( u ) = 0 .