Abstract

One of the most relevant problems about the structure of a finite group focused on the product of its conjugacy classes was posed in 1985 by Z. Arad and M. Herzog. They conjectured that in a non-abelian simple group, the product of two non-trivial classes can never be a single conjugacy class. The conjecture has been solved for several families of simple groups. We will show new results about the product of conjugacy classes regarding the non-simplicity and the normal structure of a finite group G.

Suppose that K is a conjugacy class of G. We know that KK⁻¹ is always a G-invariant set, so we can write KK⁻¹ = 1 ∪ A, where A is the join of conjugacy classes of G. When KK⁻¹ = 1 ∪ D or KK⁻¹ = 1 ∪ D ∪ D⁻¹, where D is a conjugacy class, we prove that G is not a non-abelian simple group by means of the Classification of the Finite Simple Groups (CFSG). When K is real, we also study the extreme case in which A is a single class.

(Joint work with Antonio Beltran and Maria José Felipe)