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

TL;DR

This paper investigates the structure of finite groups with specific properties of conjugacy classes of elements of prime order, establishing conditions under which the subgroup generated by such classes is soluble and describing its structure.

Contribution

It proves that if the square of a normal subset of prime order elements consists of p-elements, then the generated subgroup is soluble, extending the Baer-Suzuki theorem.

Findings

The subgroup generated by the subset is soluble.

If the group has no nontrivial p-core, then p is odd and the subgroup's structure is explicitly described.

Examples show the results are optimal.

Abstract

For $p$ a prime, $G$ a finite group and $A$ a normal subset of elements of order $p$, we prove that if $A^2 = \{ab \mid a, b \in A\}$ consists of $p$-elements then $Q = \langle A \rangle$ 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.