Inverse ambiguous functions on some finite non-abelian groups
type: Item
instance of: scholarly article
title: Inverse ambiguous functions on some finite non-abelian groups
zbMATH Open document ID: 1400.39024
author: G. Richomme
published in: Aequationes Mathematicae
publication date: 2018-09-26T00:00:00Z
review text: Let \(G\) be a group and \(N(G) =\{x \in G | x \neq x^{-1}\}\). An inverse ambiguous function on \(G\) is a bijective function \(f : G \to G\) such that \(f^{-1}(x) = (f(x))^{-1}\) for all \(x \in G\). \textit{D. J. Schmitz} [Aequationes Math. 91, No. 2, 373--389 (2017; Zbl 1372.39027)] showed that there exists an inverse ambiguous function on \(G\) if and only if \(|N(G)|\) is a multiple of \(4\). In this paper the authors employ this criterion to determine when inverse ambiguous functions exist on some common nonabelian groups, specifically the symmetric groups, the alternating groups, and the general linear groups of order \(2\) over a finite field. To count the number of elements having order greater than \(2\), they employ elementary combinatorial and group methods.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 20B30
zbMATH DE Number: 6944068
zbMATH Keywords: inverse function
zbMATH Keywords: functional equation
zbMATH Keywords: nonabelian group
MaRDI profile type: Publication
full work available at URL: s00010-018-0542-y
OpenAlex ID: W2789463783
cites work: Counting Group Elements of Order p Modulo p 2
cites work: Inverse ambiguous functions on fields
Wikidata QID: Q130135699
DOI: 10.1007/S00010-018-0542-Y
description: scientific article; zbMATH DE number 6944068