Multiplying a conjugacy class by its inverse in a finite group

Abstract

Suppose that G is a finite group and K a non-trivial conjugacy class of G such that KK −1 = 1 ∪ D ∪ D −1 with D a conjugacy class of G. We prove that G is not a non-abelian simple group. We also give arithmetical conditions on the class sizes determining the structure of K and D . Furthermore, if D = K is a non-real class, then K is p-elementary abelian for some odd prime p.

Powers of conjugacy classes in a finite group

“…then the elements of D are q-elements. In particular, the prime q must be unique, that is, q a = |O p (G) : We have seen that K n = {1} ∪ D implies that K K −1 = {1} ∪ D and this property was characterized in Theorem B of [4] in terms of characters. Thus, the hypothesis of Theorem B implies the following equality with characters.…”

“…The aim is to find some irreducible character that does not satisfy Eq. (4). Recall that the smallest integer m satisfying C m = G for each non-trivial conjugacy class C of G is called the covering number of G. The covering number of each sporadic simple group is at most 6 ( [2,13]).…”

Applying combinatorial results to products of conjugacy classes

“…In [5], the authors consider the product KK 1 for K a conjugacy class. In particular, they study [5], and they prove the following result.…”

“…In [5], the authors consider the product KK 1 for K a conjugacy class of a finite group G. In particular, they study when…”

An Arad and Fisman’s Theorem on Products of Conjugacy Classes Revisited

“…Taking one further step, several authors have studied more general conditions on the product of conjugacy classes that cannot either happen in a non-abelian simple group. This occurs, for instance, when the product of two conjugacy classes is the union of certain limited sets of conjugacy classes (see for instance [2,3,5]).…”