Groups with a solvable subgroup of prime-power index

TL;DR

This paper investigates the structure of groups containing a solvable subgroup of prime-power index, revealing how such groups relate to their solvable radical and the size constraints imposed by the index.

Contribution

It provides new insights into the properties of groups with solvable subgroups of prime-power index, including bounds on the quotient over the solvable radical.

Findings

If $G$ is non-solvable with a solvable subgroup of index $p^{eta}$, then $G/ ext{rad}(G)$ is asymptotically small compared to $p^{eta}!$.

The paper establishes properties of groups with solvable subgroups of finite prime-power index.

It characterizes the relationship between the index of solvable subgroups and the structure of the entire group.

Abstract

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index $p^{\alpha}$ (for some prime $p$), then the quotient $G/\mbox{rad}(G)$ of $G$ over the solvable radical is asymptotically small in comparison to $p^{\alpha}!$ (Theorem 4).