A nonabelian normal subgroup with a core-free projective image (Q1068195)

The following theorem of \textit{F. Menegazzo} is known [Rend. Sem. Mat. Univ. Padova 59, 11-15 (1978; reviewed above)]: If \(\pi\) is a projectivity from a finite group G to a group \(G_ 1\), \(H\triangleleft G\) and (i) G has odd order, (ii) \(H^{\pi}\) is core-free in \(G_ 1\), then H is abelian. The purpose of this paper is to answer the question of whether hypothesis (i) is necessary. The authors prove Theorem A: There are finite 2-groups G, \(G_ 1\), a normal subgroup H of G and a projectivity \(\pi\) : \(G\to G_ 1\) such that \(H^{\pi}\) is core-free in \(G_ 1\) and H is not abelian. The groups G and \(G_ 1\), constructed in the counterexample, have order \(2^{13}\) and the normal subgroup H has order \(2^ 7\). The interesting fact that there are no smaller examples is established.