A family of finite nilpotent groups (Q800481)

A finite nilpotent group \(G\) is said to be a TNP-group if for every normal subgroup \(N\) of \(G\) the transfer map \(V_{G\to N}\) of \(G\) to \(N\) has the property: \(V_{G\to N}(g) = g^{|G:N|}[N,N]\) for all \(g\in G\). The author proves that regular \(p\)-groups are TNP-groups and gives a way of constructing TNP-groups from other TNP-groups. In particular, the direct product of two TNP-groups is a TNP-group, and if \(G/Z(G)\) is a TNP-group, then the semidirect product of \(G\) and \(\operatorname{Inn}(G)\) is a TNP-group.