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

TL;DR

This paper investigates the structure of finite groups with a normal subset of odd-order elements, showing that certain closure conditions imply the subgroup generated by this subset is soluble.

Contribution

It proves that if the square of a normal subset of odd-order elements is contained in its rational closure, then the subgroup generated by this subset is soluble, providing new insights into group structure.

Findings

If $K^2 subseteq extbf D_K$, then $ extbf D_K$ contains elements of even order.

The subgroup generated by $K$ is soluble under the specified closure condition.

The result extends understanding of the expansion properties of normal subsets 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.