On the number of crossed homomorphisms
type: Item
instance of: scholarly article
title: On the number of crossed homomorphisms
zbMATH Open document ID: 0948.20016
author: Tsunenobu Asai
author: Yugen Takegahara
published in: Hokkaido Mathematical Journal
publication date: 2000-11-20T00:00:00Z
review text: The authors verify for some new classes of groups conjectures concerning the number of homomorphisms and crossed homomorphisms, respectively. These conjectures state that (H) \(|\text{Hom}(A,G)|\equiv 0\bmod\gcd(|A/A'|,|G|)\) and (I) \(|Z^1(C,H)|\equiv 0\bmod\gcd(|C|,|H|)\), where \(A\), \(G\), \(C\), \(H\) are finite groups, \(C\) is an Abelian \(p\)-group acting on the \(p\)-group \(H\) and \(Z^1(C,H)\) denotes as usual the group of 1-cocycles. Continuing work of \textit{T. Asai} and \textit{T. Yoshida} [J. Algebra 160, No. 1, 273--285 (1993; Zbl 0827.20034)] it is proved among others that (I) is true if \(H\) is an Abelian \(p\)-group, or if \(C\) is a direct product of a cyclic \(p\)-group and an elementary Abelian \(p\)-group. The last result implies that (H) holds if every Sylow subgroup of \(A/A'\) is of this type.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 1396097
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: numbers of homomorphisms
zbMATH Keywords: crossed homomorphisms
zbMATH Keywords: finite groups
zbMATH Keywords: Abelian \(p\)-groups
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1351001235
OpenAlex ID: W2027329962
DOI: 10.14492/HOKMJ/1351001235
description: scientific article; zbMATH DE number 1396097