Groups with automorphisms squaring most elements
type: Item
instance of: scholarly article
title: Groups with automorphisms squaring most elements
zbMATH Open document ID: 0665.20012
DOI: 10.1007/BF01188518
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: A universal power automorphism is an automorphism of a group G which takes any element x into \(x^ n\) for a fixed n. When \(n=-1,2\), or 3, then G is an abelian group. Many people (Liebeck, MacHale and Potter, to name a few) have asked, if an automorphism acts the same as a universal power automorphism on a fraction of the elements of a group, does this place any constraint on the structure of the group? Most of the work in this area has focused on automorphisms which invert many elements. The present paper looks at automorphisms which square many elements. The author uses elementary techniques to prove the following theorem: Let G be a finite group with an automorphism \(\theta\). Let \(S(\theta)=\{g\in G|\) \((g)\theta =g^ 2\}\) and \(r(G,\theta)=| S(\theta)| /| G|\). If \(r(G,\theta)\geq 5/12\), then \(r(G,\theta)=1,1/2\) or 5/12 and the group G is one of the following groups: (I) an abelian group of odd order and \(r(G,\theta)=1\). (II) the semidirect product, \(H\rtimes C_ 2\) where \(H=S(\theta)\) is an abelian group of odd order, \(C_ 2\) is the cyclic group of order 2 and \(r(G,\theta)=1/2\). (III) a central commutator extension \(C\rightarrowtail G\twoheadrightarrow A_ 4\) of an abelian group of odd order C, by the alternating group on 4 letters \(A_ 4\), and \(r(G,\theta)=5/12\). (A commutator extension is one in which \(G'\cap Z(G)=1.)\) Furthermore, any group G of one of these types has an automorphism \(\theta\) such that \(r(G,\theta)\) has the value given above.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4087908
zbMATH Keywords: universal power automorphism
zbMATH Keywords: finite group
zbMATH Keywords: abelian group of odd order
zbMATH Keywords: semidirect product
zbMATH Keywords: central commutator extension
author: Jay Zimmerman
reviewed by: Jay Zimmerman
MaRDI profile type: Publication
cites work: Q3739341
cites work: Groups with an automorphism squaring many elements
cites work: Nonsolvable groups with an automorphism inverting many elements
cites work: Q3882615
cites work: Homology in group theory
description: scientific article; zbMATH DE number 4087908