On the number of solutions of equation \(x^{p^ k}=1\) in a finite group
type: Item
instance of: scholarly article
title: On the number of solutions of equation \(x^{p^ k}=1\) in a finite group
zbMATH Open document ID: 0840.20017
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1996-06-30T00:00:00Z
full work available at URL: https://eudml.org/doc/244269
review text: Let \(G\) be a finite group, \(N(t, G)\) is the number of solutions of the equation \(x^t=1\) in \(G\). Conditions under which \(N(p^k, G)\) in a finite \(p\)-group is divisible by \(p^{n+k}\) (here \(n\) is a fixed positive integer) are given. It is shown also that congruences for the number of cyclic subgroups of order \(p^k\) which are true for abelian groups hold for more general finite groups (in particular for groups with abelian Sylow \(p\)-subgroups).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 817257
zbMATH Keywords: number of solutions
zbMATH Keywords: congruences
zbMATH Keywords: number of cyclic subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: groups with Abelian Sylow \(p\)-subgroups
author: Yakov Berkovich
reviewed by: N. M. Kurnosenko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 817257