Squares of conjugacy classes and a variant on the Baer–Suzuki Theorem
摘要
For p a prime, G a finite group and A a normal subset of elements of order p, we prove that if A2 = {ab ∣ a,b ∈ A} consists of p-elements then Q = 〈A〉 is soluble. Further, if Op(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.