On the embedding of core-free projective images of normal subgroups
Given a projectivity (i.e. a subgroup lattice isomorphism) between two groups G and \(\bar G,\) we adopt the convention that subgroups which correspond under the projectivity are denoted by the same letter, e.g. X and \(\bar X.\) It is known that \(H\triangleleft G\) does not ensure \(\bar H\triangleleft \bar G\); in fact we assume throughout that (*) \(H\triangleleft G\) but \(\bar H\) is core-free in \(\bar G.\) Now suppose that G/H is finitely generated and \(\bar H\) is quasinormal (permutable) in \(\bar G.\) By a result of Zacher, \(\bar H\) is periodic nilpotent, so for a fixed prime p we take \(\bar S\) to be the Sylow p- subgroup of \(\bar H.\) Write \(\bar C\) for the centralizer in \(\bar G\) of the normal closure \(\bar N\) of \(\bar S\) in \(\bar G,\) and D for the centralizer in G of the corresponding N. The author proves that \(\bar G/\bar C\) and \(G/D\) are p-groups (known results force \(N\triangleleft G).\) The second main theorem in the paper generalizes a result of \textit{R. Schmidt} [Proc. Lond. Math. Soc., III. Ser. 30, 287-300 (1975; Zbl 0299.20015)]. Schmidt proved that if (*) is assumed and G is finite then there is a \(\bar G\)-invariant series, with cyclic factors, in the normal closure \(\bar M\) of \(\bar H\) in \(\bar G,\) and a G-invariant series, also with cyclic factors, in the corresponding M; moreover if \(\bar H\) is quasinormal in \(\bar G\) then the factors in these series are centralized by \(\bar G,\) G respectively. In the present paper it is shown that ''G finite'' can be weakened to ''every periodic quotient of a finitely- generated subgroup of G is finite'', and the same conclusions hold. The proofs are relatively straightforward, - the notation is not.
scientific article; zbMATH DE number 3953066
scientific article