Double chain conditions in infinite groups Mattia Brescia - PowerPoint PPT Presentation

Double chain conditions in infinite groups Mattia Brescia Universit` a degli Studi di Napoli Federico II Gruppen und topologische Gruppen 2017 Trento – 16 June, 2017 Mattia Brescia Double chain conditions in infinite groups 1 / 16

Finiteness Conditions Let U be the universe of all groups and let F be the class of finite groups. Any intermediate class between them, i.e. any X such that F � X � U , is said to be a finiteness class. The property of belonging to such class is called a finiteness or finitary condition. Classical non-trivial examples of finiteness conditions are Locally finiteness; Periodicity; Hopficity; ... Mattia Brescia Double chain conditions in infinite groups 2 / 16

Finiteness Conditions Let U be the universe of all groups and let F be the class of finite groups. Any intermediate class between them, i.e. any X such that F � X � U , is said to be a finiteness class. The property of belonging to such class is called a finiteness or finitary condition. Classical non-trivial examples of finiteness conditions are Locally finiteness; Periodicity; Hopficity; ... Mattia Brescia Double chain conditions in infinite groups 2 / 16

Chain Conditions Let T be a totally ordered set and χ be a group theoretical prop- erty. We will say that a group G satisfies the T -chain condition on χ -subgroups if there is no increasing function between T and the set of the χ -subgroups of G ordered by inclusion. Such class will be called a chain class and will here be denoted by C ( T , χ ) . Very known examples of chain classes are For T = ( N , < ) and χ = ”to be a group”, C ( T ; χ ) = Max ; For T = ( N , > ) and χ = ”to be a group”, C ( T ; χ ) = Min ; For T = ( N , < ) and χ = ”to be a finite group”, C ( T ; χ ) = Max - f ... Mattia Brescia Double chain conditions in infinite groups 3 / 16

Chain Conditions Let T be a totally ordered set and χ be a group theoretical prop- erty. We will say that a group G satisfies the T -chain condition on χ -subgroups if there is no increasing function between T and the set of the χ -subgroups of G ordered by inclusion. Such class will be called a chain class and will here be denoted by C ( T , χ ) . Very known examples of chain classes are For T = ( N , < ) and χ = ”to be a group”, C ( T ; χ ) = Max ; For T = ( N , > ) and χ = ”to be a group”, C ( T ; χ ) = Min ; For T = ( N , < ) and χ = ”to be a finite group”, C ( T ; χ ) = Max - f ... Mattia Brescia Double chain conditions in infinite groups 3 / 16

Some history First examples coming in were Noetherian groups (R. Baer, K. A. Hirsch, O. H. Kegel, ...); Artinian groups (S. N. ˇ Cernikov, V. P. ˇ Sunkov, D. I. Zaicev, ...); Max-n groups (P. Hall, D. H. McLain, J. S. Wilson, ...); Max-ab groups (A. I. Malˇ cev, B. I. Plotkin, O. J. Schmidt, ...). Mattia Brescia Double chain conditions in infinite groups 4 / 16

Aiming for cool results Some early relevant questions were Is Max = ( P C ) F ? Is Min = ˇ C? Is Max - sn = Max in the universe of locally soluble groups? Mattia Brescia Double chain conditions in infinite groups 5 / 16

Ending up with monsters The answers to which are sanski˘ Is Max = ( P C ) F ? NO. (Thank you, Ol’ˇ i) Is Min = ˇ sanski˘ C? NO. (Thank you again, Ol’ˇ i) Is Max - sn = Max in the universe of locally soluble groups? Who knows! Mattia Brescia Double chain conditions in infinite groups 6 / 16

Ending up with monsters The answers to which are sanski˘ Is Max = ( P C ) F ? NO. (Thank you, Ol’ˇ i) Is Min = ˇ sanski˘ C? NO. (Thank you again, Ol’ˇ i) Is Max - sn = Max in the universe of locally soluble groups? Who knows! Mattia Brescia Double chain conditions in infinite groups 6 / 16

Ending up with monsters The answers to which are sanski˘ Is Max = ( P C ) F ? NO. (Thank you, Ol’ˇ i) Is Min = ˇ sanski˘ C? NO. (Thank you again, Ol’ˇ i) Is Max - sn = Max in the universe of locally soluble groups? Who knows! Mattia Brescia Double chain conditions in infinite groups 6 / 16

Changing the point of view For many years the quasi-totality of investigations about chain conditions were about changing the group theoretical property χ and showing results about Max- χ or Min- χ , i.e. about the classes of the groups satisfying the maximal or the minimal condition on χ -subgroups. A recent inspiration came reading the works of D. I. Zaicev and T. S. Shores who, in particular, studied the class of C ( Z ; U ) , where U is the trivial universal property. D. I. Zaicev (1971), T. S. Shores (1973) - Let G be a locally radical group. Then G satisfies the double chain condition on subgroups i ff it satisfies either the maximal or the minimal condition on subgroups. Mattia Brescia Double chain conditions in infinite groups 7 / 16

Changing the point of view For many years the quasi-totality of investigations about chain conditions were about changing the group theoretical property χ and showing results about Max- χ or Min- χ , i.e. about the classes of the groups satisfying the maximal or the minimal condition on χ -subgroups. A recent inspiration came reading the works of D. I. Zaicev and T. S. Shores who, in particular, studied the class of C ( Z ; U ) , where U is the trivial universal property. D. I. Zaicev (1971), T. S. Shores (1973) - Let G be a locally radical group. Then G satisfies the double chain condition on subgroups i ff it satisfies either the maximal or the minimal condition on subgroups. Mattia Brescia Double chain conditions in infinite groups 7 / 16

Changing the point of view So in 2005 there came the first new work on the so-called ”Double chain condition”. F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property of ”being normal”. Then, in the universe of residually finite groups, DC n = Max- n . F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property of ”being normal”. Then, in the universe of periodic soluble groups, DC n = Min- n . Mattia Brescia Double chain conditions in infinite groups 8 / 16

Changing the point of view So in 2005 there came the first new work on the so-called ”Double chain condition”. F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property of ”being normal”. Then, in the universe of residually finite groups, DC n = Max- n . F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property of ”being normal”. Then, in the universe of periodic soluble groups, DC n = Min- n . Mattia Brescia Double chain conditions in infinite groups 8 / 16

Changing the point of view Moreover in the same paper F. De Mari, F. de Giovanni (2005) - Let nn be the subgroup property of ”being not normal”. Then, in the universe of locally radical groups, G is a DC nn -group if and only if G satisfies either Max- nn or Min- nn . So is everything this predictable? Mattia Brescia Double chain conditions in infinite groups 9 / 16

Changing the point of view Moreover in the same paper F. De Mari, F. de Giovanni (2005) - Let nn be the subgroup property of ”being not normal”. Then, in the universe of locally radical groups, G is a DC nn -group if and only if G satisfies either Max- nn or Min- nn . So is everything this predictable? Mattia Brescia Double chain conditions in infinite groups 9 / 16

Going on with the ordering type of Z Theorem F. de Giovanni, M. B. – 2015 Let G be a radical group. G satisfies DC sn if and only if G satisfies one of the following: G satisfies Max- sn ; G satisfies Min- sn ; G = HJ where J is the finite residual of G , H is polycyclic, C H ( J ) is finite and every subnormal subgroup of G is either Min or properly contains J . Not everything is this predictable, indeed! Mattia Brescia Double chain conditions in infinite groups 10 / 16

Going on with the ordering type of Z Theorem F. de Giovanni, M. B. – 2015 Let G be a radical group. G satisfies DC sn if and only if G satisfies one of the following: G satisfies Max- sn ; G satisfies Min- sn ; G = HJ where J is the finite residual of G , H is polycyclic, C H ( J ) is finite and every subnormal subgroup of G is either Min or properly contains J . Not everything is this predictable, indeed! Mattia Brescia Double chain conditions in infinite groups 10 / 16

Going on with the ordering type of Z Theorem F. de Giovanni, M. B. – 2015 Let G be a radical group. G satisfies DC sn if and only if G satisfies one of the following: G satisfies Max- sn ; G satisfies Min- sn ; G = HJ where J is the finite residual of G , H is polycyclic, C H ( J ) is finite and every subnormal subgroup of G is either Min or properly contains J . Not everything is this predictable, indeed! Mattia Brescia Double chain conditions in infinite groups 10 / 16

Double chains vs non-subnormal subgroups M.B. (2016) - Let sn be the subgroup property of ”being not sub- normal”. Then, in the universe of infinite locally finite groups, DC sn = ˇ C ∪ S , where S is the class of groups with every sub- group subnormal. Mattia Brescia Double chain conditions in infinite groups 11 / 16

Double chains vs snn-subgroups M.B. (2016) - Let G be a finitely generated soluble DC snn -group. Then G satisfies Max. M.B. (2016) - Let G be a Baer DC snn -group. Then G is nilpotent. In particular, G satisfies either Max- nn or Min- nn . Mattia Brescia Double chain conditions in infinite groups 12 / 16