Lifting in Frattini Covers and A Characterization of Finite Solvable Groups

2011/12/19 by Robert M. Guralnick, Robert Guralnick, Guralnick, Robert +2
Mathematics · #14H30 (Secondary) #20D05 (Primary) 14G32 #20D06 #20D10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Representation Theory (math.RT) #math.AG #math.GR #math.RT #msc:14G32 #msc:14H30 #msc:20D05 #msc:20D06 #msc:20D10

paper · pdf · doi:10.48550/arxiv.1112.4559

openalex publication_date 2011/12/19 · arxiv created 2011/12/20 · arxiv updated 2011/12/21 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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.

Related

• Hurwitz spaces and liftings to the Valentiner group
• Hurwitz space components; and the Coleman-Oort Conjecture
• On the orders of the non-Frattini elements of a finite group
• Orbifold groups, quasi-projectivity and covers
• Lifting Elementary Abelian Covers of Curves