On n-isoclinic embedding of groups
Two groups G and H are called n-isoclinic if there exists an isomorphism \(\alpha\) of \(G/Z_ n(G)\) onto \(H/Z_ n(H)\) that induces an isomorphism \(\beta\) of \(\gamma_ n(G)\) onto \(\gamma_ n(H)\) in the natural way. It was shown by \textit{N. S. Hekster} [J. Pure Appl. Algebra 40, 63-85 (1986; Zbl 0593.20040)] that 1-isoclinic groups G and H can be embedded in a group X such that \(G\cdot Z_ 1(X)=X=H\cdot Z_ 1(X)\). Following suggestions of F. R. Beyl and J. Tappe, the author deduces this result from a diagram on p. 130 in \textit{F. R. Beyl} and \textit{J. Tappe}, Group extensions, Representations, and the Schur Multiplicator [Lect. Notes Math. 958 (1982; Zbl 0544.20001)]. It is not clear whether there is an analogue to the above for n-isoclinic groups, however, the following somewhat weaker result is proved: Theorem. If G and H are n-isoclinic, then \(U=G/[G,Z_ n(G)]\) and \(V=H/([G,Z_ n(G)]\cap \gamma_{n+1}(G))^{\beta}\) are n-isoclinic and there exists a group X with subgroups \(\bar U\cong U\), \(\bar V\cong V\) such that \(\bar U\cdot Z_ n(X)=X=\bar V\cdot Z_ n(X)\). Moreover, U and V are n-isoclinic to X.
scientific article; zbMATH DE number 4055891
scientific article