Lifting in Frattini Covers and A Characterization of Finite Solvable Groups

TL;DR

This paper establishes a lifting theorem for odd Frattini covers of finite groups and characterizes solvable and p-solvable groups through element triples, connecting to modular tower and Hurwitz space properties.

Contribution

It introduces a new lifting theorem for odd Frattini covers and provides a novel characterization of solvable groups via element triples.

Findings

Lifting theorem for odd Frattini covers proved

Characterization of solvable groups using element triples

Connections to modular tower and Hurwitz space properties

Abstract

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1. This is also related to various questions about Fried's modular tower program and properties of Hurwitz spaces of covers of curves.