Groups with automorphisms squaring most elements
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.
scientific article; zbMATH DE number 4087908