Structure of normal subgroups with three G-class sizes

Abstract

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Solvability of normal subgroups and $G$-class sizes

“…In view of Camina's work, the solution of this problem seems more difficult and it needs different techniques from those employed in [1]. In this paper, we show the following contribution to our conjecture.…”

“…This can only occur at most for the groups that belong to the following list (see Chapter 5 of [14]). i) Certain linear groups: M (A 1 (3 2 )) of order 6; M (A 2 (2 2 )) of order 48; and M (A n (q)) for (n, q) = (1, 4), (1,9), (2,2), (2,4), (3,2), that has order gcd(n + 1, q − 1). In the first two cases the corresponding group does not have conjugacy classes of size dividing 6 and 48, respectively.…”

“…Suppose that cs G (N ) = {1, m, n} with 1 < m < n. We observe that if m does not divide n then Theorem C of [1] implies that N is solvable. Hence, as N is nonabelian, we have F(N/Z(N )) > 1 and the result follows.…”

“…Normal subgroups having exactly two G-class sizes have been proved to be nilpotent [3]. Also, in [1], the structure of a normal subgroup with three G-class sizes, under certain arithmetical conditions on these sizes, is determined. The proof is inspired by a generalization for normal subgroups of the concept of F -group.…”

Graphs associated to conjugacy classes of normal subgroups in finite groups

“…In addition, Theorem E extends Corollary B of [2], which analyzes (with a completely different approach) the particular case in which N has exactly two coprime G-class sizes bigger than 1.…”