Normal subgroups and projectivities (Q1076799)

If \(\phi\) is a projectivity from the group G onto a group \(\bar G \)(i.e., an isomorphism from the subgroup lattice of G to that of \(\bar G)\) and \(N\trianglelefteq G\), then \(N^{\phi}\) need not be normal in \(\bar G.\) Therefore it is natural to consider the core \(K^{\phi}\) and the normal closure \(H^{\phi}\) of \(N^{\phi}\) in \(\bar G.\) It is well known that H and K are normal subgroups of G and the deviation of \(N^{\phi}\) from normality is described by the structure of H/K, H/N, N/K and of the projective images of these groups. In the paper under review it is shown that N/K and \(N^{\phi}/K^{\phi}\) are solvable groups of derived length at most 3 and 4, respectively; in a subsequent note by the author and \textit{F. Menegazzo} [Rend. Semin. Mat. Univ. Padova 73, 249-260 (1985; reviewed below)] the bound on the derived length of \(N^{\phi}/K^{\phi}\) has been improved to 3. The first step in the direction of this result was made by \textit{F. Menegazzo} [ibid. 59, 11-15 (1979; Zbl 0581.20018)] who proved that for finite groups of odd order N/K is abelian. Using Zacher's theorem that the finiteness of the index of a subgroup is invariant under projectivities it is not difficult to see that as in Menegazzo's paper the critical situation in the proof of the author's more general theorem is that in which G and \(\bar G\) are finite p-groups with G/N cyclic and \(N^{\phi}\) core-free in \(\bar G,\) and by Menegazzo's result it remained to consider the case \(p=2\). For this case, the author and \textit{S. E. Stonehewer} [J. Algebra 97, 329-346 (1985; Zbl 0581.20019)] have given an example in which N and \(N^{\phi}\) have derived length 2. The main lemma of the paper under review states that N in fact is abelian if \(| N/\Omega_{n-1}(N)| \geq 8\), where \(2^ n=Exp N\). Then if \(s=\min \{r:\) \(| \Omega (N/\Omega_ r(N))| \leq 4\}\) it is not hard to show that \(N/\Omega_ s(N)\) is metacyclic and \(\Omega_ s(N)\) is abelian, and the result follows. It is not known whether it is best possible; the exact bound on the derived length of N/K and \(N^{\phi}/K^{\phi}\) might be 2. To prove the main lemma the author finds generators and relations for large parts of N and uses these very cleverly to show that N has modular subgroup lattice and, finally, is abelian. This long and complicated proof is well organized and the paper is clearly written and readable.