International Journal of Group Theory
Volume:7 Issue: 1, Mar 2018

The structure of infinite groups in which every (proper) normal subgroup is the only one of its cardinality is investigated in the universe of groups without infinite simple sectionsý. ýThe corrisponding problem for finite soluble groups was considered by Mý. ýCurzio (1958)ý.

A group class X is said to be countably recognizable if a group belongs to X whenever all its countable subgroups lie in X ý. ýIt is proved here that most of the relevant classes of groups defined by restrictions on the conjugacy classes are countably recognizableý.

We prove that every finite group G contains a three-generated subgroup H with the following propertyý: ýa prime p divides the degree of an irreducible character of G if and only if it divides the degree of an irreducible character of H. There is no analogous result for the prime divisors of the sizes of the conjugacy classesý.

We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an influence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau's Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions.

This paper deals with the mutual relationships between the factor group G/ζ(G) (respectively G/ζ k (G)) and G ′ (respectively γ k (G) and G N). It is proved that if G/ζ(G) (respectively G/ζ k (G)) has finite 0 -rank, then G ′ (respectively γ k (G) and G N) also have finite 0 -rank. Furthermore, bounds for the 0 -ranks of G ′ ,γ k (G) and G N are obtained.

ýIn this paper we highlight the connection between certain classes of regular subgroups of the affine groupý ýAGL n (F) ý, ýF a fieldý, ýand associative nilpotent F -algebras of dimension ný. ýWe also describe how the classification of projective congruence classes of square matrices is equivalent to theý ýclassification of regular subgroups of particular shapeý.

The structure of groups which have at most two isomorphism classes of central factors (B 2-groups) are investigatedý. ýA complete description of B 2-groups is obtained in the locally finite case and in the nilpotent caseý. ýIn addition detailedinformation is obtained about soluble B 2-groupsý. ýAlso structural information about insoluble B 2-groups is givený, ýin particular when such a group has the derived subgroup satisfying the minimal conditioný.