Non-solvable groups all of whose indices are odd-square-free

Abstract

Given a finite group G and x is an element of G, the class size of x in G is called odd-square-free if it is not divisible by the square of any odd prime number. In this paper, we show that if G is a nonsolvable finite group, all of whose class sizes are odd-square-free, then we have some control on the structure of G, which is an answer to the dual of the question mentioned by Huppert in [5].