Groups in which every proper subgroup of Infinite Rank is Finite Rank-by-Hypercentral or Hypercentral-by-Finite Rank PRESENTED BY : ZITOUNI AMEL SUPERVISOR :PR. TRABELSI NADIR UNIVERSITY OF SETIF 1 - ALGERIA YRAC 2017 NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 1 / 31 Richard Brauer " A tremendous effort has been made by mathematicians for more than a century to clear up the chaos in group theory. Still, we cannot answer some of the simplest questions." NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 2 / 31 2 If N is a normal subgroup of an -group G, then at least one of X the groups N and G/N belongs to . X 3 No finite cyclic group lies in . X Class of large groups Following the context of M. de Falco – F. de Giovanni – C. Musella, (2014), the notion of a “large group” can be formalized in the following way : Definition Let be a class of groups. Then is said to be a class of large groups if itX satisfies the following conditionsX : 1 If a group G contains an -subgroup, then G belongs to . X X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 3 / 31 3 No finite cyclic group lies in . X Class of large groups Following the context of M. de Falco – F. de Giovanni – C. Musella, (2014), the notion of a “large group” can be formalized in the following way : Definition Let be a class of groups. Then is said to be a class of large groups if itX satisfies the following conditionsX : 1 If a group G contains an -subgroup, then G belongs to . X X 2 If N is a normal subgroup of an -group G, then at least one of X the groups N and G/N belongs to . X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 3 / 31 Class of large groups Following the context of M. de Falco – F. de Giovanni – C. Musella, (2014), the notion of a “large group” can be formalized in the following way : Definition Let be a class of groups. Then is said to be a class of large groups if itX satisfies the following conditionsX : 1 If a group G contains an -subgroup, then G belongs to . X X 2 If N is a normal subgroup of an -group G, then at least one of X the groups N and G/N belongs to . X 3 No finite cyclic group lies in . X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 3 / 31 The rank of a group Definition A group is said to have finite rank r if every finitely generated subgroup can be generated by r elements, and r is the least positive integer with this property. If no such integer r exists then we say that the group has infinite rank. Groups of infinite rank form a class of large groups. NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 4 / 31 Every polycyclic group and every Cernikovˇ group has finite rank. A soluble minimax group has finite rank. Any free non-abelian has infinite rank. Examples Examples Cp¥ , Q are locally cyclic groups, so they have rank 1. NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 5 / 31 A soluble minimax group has finite rank. Any free non-abelian has infinite rank. Examples Examples Cp¥ , Q are locally cyclic groups, so they have rank 1. Every polycyclic group and every Cernikovˇ group has finite rank. NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 5 / 31 Any free non-abelian has infinite rank. Examples Examples Cp¥ , Q are locally cyclic groups, so they have rank 1. Every polycyclic group and every Cernikovˇ group has finite rank. A soluble minimax group has finite rank. NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 5 / 31 Examples Examples Cp¥ , Q are locally cyclic groups, so they have rank 1. Every polycyclic group and every Cernikovˇ group has finite rank. A soluble minimax group has finite rank. Any free non-abelian has infinite rank. NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 5 / 31 Large class controls a subgroup property Let be a class of large groups, and let be a property pertainingX to subgroups of a group. P Definition controls if and only if the following condition is satisfied : X P G is any -group and all -subgroups of G have the property , then Xholds for all subgroupsX of G. P P NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 6 / 31 Examples Examples The class of cyclic groups controls periodicity. The class of finitely generated groups controls commutativity. NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 7 / 31 G R if for each 1 = x G there is a normal subgroup N 2 X 6 2 x of G such that x / Nx and G/Nx . G L if every2 finite subset of G2is X contained in an -subgroup.2 X GX P´ if G has an ascending series each of whose factors is a 2-group.X GX P` if G has a descending series each of whose factors is a 2-group.X X Let L denote the set of closure operations R; L; P´ ; P` . If is any class of groups, then : f g X Strongly locally graded groups A group G is locally graded if every finitely generated nontrivial subgroup of G has a finite nontrivial image. NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 8 / 31 G R if for each 1 = x G there is a normal subgroup N 2 X 6 2 x of G such that x / Nx and G/Nx . G L if every2 finite subset of G2is X contained in an -subgroup.2 X GX P´ if G has an ascending series each of whose factors is a 2-group.X GX P` if G has a descending series each of whose factors is a 2-group.X X Strongly locally graded groups A group G is locally graded if every finitely generated nontrivial subgroup of G has a finite nontrivial image. Let L denote the set of closure operations R; L; P´ ; P` . If is any class of groups, then : f g X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 8 / 31 G L if every finite subset of G is contained in an -subgroup.2 X GX P´ if G has an ascending series each of whose factors is a 2-group.X GX P` if G has a descending series each of whose factors is a 2-group.X X Strongly locally graded groups A group G is locally graded if every finitely generated nontrivial subgroup of G has a finite nontrivial image. Let L denote the set of closure operations R; L; P´ ; P` . If is any class of groups, then : f g X G R if for each 1 = x G there is a normal subgroup Nx of 2G suchX that x / N 6 and2G/N . 2 x x 2 X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 8 / 31 G P´ if G has an ascending series each of whose factors is a 2-group.X GX P` if G has a descending series each of whose factors is a 2-group.X X Strongly locally graded groups A group G is locally graded if every finitely generated nontrivial subgroup of G has a finite nontrivial image. Let L denote the set of closure operations R; L; P´ ; P` . If is any class of groups, then : f g X G R if for each 1 = x G there is a normal subgroup N 2 X 6 2 x of G such that x / Nx and G/Nx . G L if every2 finite subset of G2is X contained in an -subgroup.2 X X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 8 / 31 G P` if G has a descending series each of whose factors is a 2-group.X X Strongly locally graded groups A group G is locally graded if every finitely generated nontrivial subgroup of G has a finite nontrivial image. Let L denote the set of closure operations R; L; P´ ; P` . If is any class of groups, then : f g X G R if for each 1 = x G there is a normal subgroup N 2 X 6 2 x of G such that x / Nx and G/Nx . G L if every2 finite subset of G2is X contained in an -subgroup.2 X GX P´ if G has an ascending series each of whose factors is a 2-group.X X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 8 / 31 Strongly locally graded groups A group G is locally graded if every finitely generated nontrivial subgroup of G has a finite nontrivial image. Let L denote the set of closure operations R; L; P´ ; P` . If is any class of groups, then : f g X G R if for each 1 = x G there is a normal subgroup N 2 X 6 2 x of G such that x / Nx and G/Nx . G L if every2 finite subset of G2is X contained in an -subgroup.2 X GX P´ if G has an ascending series each of whose factors is a 2-group.X GX P` if G has a descending series each of whose factors is a 2-group.X X NAPLI, 24 TH 2017 (UNIVERSITY OF SETIF 1 - ALGERIA ) YRAC 2017 8 / 31 In 1990, N.