On the behaviour of normal subgroups under projectivities
Let G and \(G_ 1\) be groups. A projectivity \(\pi\) : \(G\to G_ 1\) is an isomorphism from the subgroup lattice of G onto the subgroup lattice of \(G_ 1\). If H is a normal subgroup if G, it is well-known that the preimage N of the core of \(H^{\pi}\) in \(G_ 1\) is a normal subgroup of G, and the knowledge of the factor groups H/N and \(H^{\pi}/N^{\pi}\) is useful to understand the connections between the structures of G and \(G_ 1.\) Here the authors prove the following: Let \(G=H(a)\) be a finite 2-group, where H is normal in G, and let \(\pi\) : \(G\to G_ 1\) be a projectivity such that the core of \(H^{\pi}\) in \(G_ 1\) is trivial. Then \(H'\) has order at most 2 and H is a modular group. Moreover, G is metabelian and \(G_ 1\) has derived length at most 3. - From this they deduce the following general results: (1) Let \(\pi\) : \(G\to G_ 1\) be a projectivity, H a normal subgroup of G and N the preimage of the core of \(H^{\pi}\) in \(G_ 1\). Then H/N and \(H^{\pi}/N^{\pi}\) are residually (finite modular) groups; in particular they are metabelian. Moreover H/N is nilpotent of class at most 2 and has commutator subgroup of exponent at most 2. (2) Let \(\pi\) : \(G\to G_ 1\) be a projectivity, and let X be a subgroup of G. If the n-th commutator subgroup of G is contained in X, then the 3n-th commutator subgroup of \(G_ 1\) is contained in \(X^{\pi}\). In particular, if G is soluble with derived length n, then \(G_ 1\) is soluble with derived length at most 3n-1. These theorems improve previous results of \textit{G. Busetto} [J. Algebra 88, 52-62 (1984; Zbl 0594.20021)] and \textit{G. Busetto} and \textit{F. Menegazzo} [Rend. Semin. Mat. Univ. Padova 73, 249-260 (1985; Zbl 0594.20022)].
scientific article; zbMATH DE number 4210402
scientific article