Expansion of normal subsets of odd-order elements in finite groups

Abstract

Let $G$ be a finite group and $K$ a normal subset consisting of odd-order elements. The rational closure of $K$, denoted $\mathbf D_K$, is the set of elements $x \in G$ with the property that $\langle x \rangle = \langle y \rangle$ for some $y$ in $K$. If $K^2 \subseteq \mathbf D_K$, we prove that $\langle K \rangle$ is soluble.