Squares of conjugacy classes and a variant on the Baer–Suzuki Theorem

Abstract For p a prime, G a finite group and A a normal subset of elements of order p , we prove that if A 2 = { ab ∣ a , b ∈ A } consists of p -elements then Q = 〈 A 〉 is soluble. Further, if O p ( G ) = 1, we show that p is odd, F ( Q ) is a non-trivial p ′-group and Q/F ( Q ) is an elementary abelian p -group. We also provide examples which show this conclusion is best possible.