On the number of crossed homomorphisms
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.
scientific article; zbMATH DE number 1396097