A Note On Transversals

TL;DR

This paper explores conditions under which the solvability of a finite group can be deduced from the properties of a generating transversal of a core-free subgroup, revealing new structural insights.

Contribution

It establishes that the existence of a solvable, generating transversal implies the group is solvable, and characterizes the subgroup H as elementary abelian 2-group under certain conditions.

Findings

Solvable, generating transversals imply the group is solvable.

If the transversal has a specific invariant sub right loop, then H is elementary abelian 2-group.

Provides new criteria linking transversals to group solvability and subgroup structure.

Abstract

Let $G$ be a finite group and $H$ a core-free subgroup of $G$. We will show that if there exists a solvable, generating transversal of $H$ in $G$, then $G$ is a solvable group. Further, if $S$ is a generating transversal of $H$ in $G$ and $S$ has order 2 invariant sub right loop $T$ such that the quotient $S/T$ is a group. Then $H$ is an elementary abelian 2-group.