Noninner automorphisms of order \(p\) of finite \(p\)-groups
type: Item
instance of: scholarly article
title: Noninner automorphisms of order \(p\) of finite \(p\)-groups
zbMATH Open document ID: 1012.20017
author: Marian Deaconescu
author: Gheorghe Silberberg
published in: Journal of Algebra
publication date: 2003-03-13T00:00:00Z
review text: The central result of the paper is the following theorem: Let \(G\) a finite nonabelian \(p\)-group such that \(\Phi(G)\neq C_G(Z(\Phi(G)))\). Then \(G\) has a noninner automorphism which fixes \(\Phi(G)\). This result reduces the verification of the conjecture that every finite nonabelian \(p\)-group has a noninner automorphism of order \(p\) to the case when \(\Phi(G)=C_G(Z(\Phi(G)))\). This conjecture is the sharpened version of the well-known theorem of Gaschütz asserting that any finite nonabelien \(p\)-group has a noninner automorphism of \(p\)-power order.
reviewed by: Vahagn H. Mikaelian
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1762100
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: noninner automorphisms of order \(p\)
zbMATH Keywords: Frattini subgroup
MaRDI profile type: Publication
full work available at URL: jabr.2001.9093
OpenAlex ID: W2053744932
Wikidata QID: Q56094146
cites work: Nichtabelsche p-Gruppen besitzen äußere p-Automorphismen
cites work: Q5577154
cites work: Endliche Gruppen I
cites work: Outer Automorphisms in Nilpotent p -Groups of Class 2
cites work: Frattinian \(p\)-groups
cites work: A cohomological property of regular \(p\)-groups
DOI: 10.1006/JABR.2001.9093
description: scientific article; zbMATH DE number 1762100