On the number of crossed homomorphisms (Q1962713)

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.