Groups with finite conjugacy classes of non-subnormal subgroups

The structure of groups for which certain sets of commutator subgroups are finite is investigated. In particular, the relation between such groups and groups with finite conjugacy classes of elements is discussed.

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INFINITE GROUPS WITH TWO CONJUGACY CLASSES OF NON-SUBNORMAL SUBGROUPS

JP Journal of Algebra Number Theory and Applications ◽

10.17654/nt038050489 ◽

2016 ◽

Vol 38 (5) ◽

pp. 489-497

Author(s):

Aifang Feng ◽

Zuhua Liu

Keyword(s):

Conjugacy Classes ◽

Infinite Groups ◽

Subnormal Subgroups

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Maximal subgroups of GL n (D) with finite conjugacy classes

manuscripta mathematica ◽

10.1007/s00229-009-0290-3 ◽

2009 ◽

Vol 130 (3) ◽

pp. 287-293 ◽

Cited By ~ 5

Author(s):

Dariush Kiani ◽

Mojtaba Ramezan-Nassab

Keyword(s):

Conjugacy Classes ◽

Maximal Subgroups ◽

Finite Conjugacy

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The Dietzmann property of some classes of groups with locally finite conjugacy classes

Journal of Algebra ◽

10.1016/j.jalgebra.2004.01.026 ◽

2004 ◽

Vol 277 (1) ◽

pp. 364-369

Author(s):

Rudolf Maier

Keyword(s):

Conjugacy Classes ◽

Locally Finite ◽

Finite Conjugacy

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Weak Cayley tables and generalized centralizer rings of finite groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004112000205 ◽

2012 ◽

Vol 153 (2) ◽

pp. 281-318 ◽

Cited By ~ 7

Author(s):

STEPHEN P. HUMPHRIES ◽

EMMA L. RODE

Keyword(s):

Finite Group ◽

Finite Groups ◽

Conjugacy Classes ◽

Cayley Table ◽

A Finite Group ◽

Finite Conjugacy ◽

Relationship Of ◽

Cayley Tables ◽

The Relationship

AbstractFor a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

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A NILPOTENCY-LIKE CONDITION FOR INFINITE GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788717000416 ◽

2018 ◽

Vol 105 (1) ◽

pp. 24-33

Author(s):

M. DE FALCO ◽

F. DE GIOVANNI ◽

C. MUSELLA ◽

N. TRABELSI

Keyword(s):

Normal Subgroup ◽

Positive Integer ◽

Finite Index ◽

Conjugacy Classes ◽

Nilpotent Subgroup ◽

Infinite Groups ◽

Finite Conjugacy

If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$-property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$. Thus, $FE_{1}$-groups are precisely those groups with finite conjugacy classes ($FC$-groups) and the aim of this paper is to extend properties of $FC$-groups to the case of groups with the $FE_{k}$-property for $k>1$. The class of $FE_{k}$-groups contains the relevant subclass $FE_{k}^{\ast }$, consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$, whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$.

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