Connected transversals to subnormal subgroups. (English) Zbl 0889.20020

Let \(H\) be a subgroup of \(G\) and denote by \(L_G(H)=\bigcap H^g\), the core of \(H\). Define normal subgroups \(Z_{H,n}(G)\) and \(Z^*_{H,n}(G)\) by \(Z_{H,0}(G)=Z_{H,0}^*=L_G(H)\), \(Z_{H,n}(G)\subseteq Z_{H,n+1}^*(G)\), \(Z_{H,n+1}(G)/Z_{H,n}(G)=Z(G/Z_{H,n}(G))\) and \(Z_{H,n+1}(G)=L_G(HZ^*_{H,n+1}(G))\).
The paper contains a number of basic properties of these series of subgroups, and then the situation is considered when there exist transversals \(A\) and \(B\) of \(H\) with \([A,B]\subseteq G\). (This condition is connected to motivation coming from quasigroups, a fact that is mentioned but not elaborated.) In that case \(HZ^*_{H,n}(G)\) equals \(N_{G,n}(H)\), where \(n\) stands for the \(n\)-th iterate of the normalizer, and \(Z_{H,n}(G)\) equals \(L_G(N_{G,n}(H))\). Some conclusions are drawn out of these equalities and several examples are added.


20F14 Derived series, central series, and generalizations for groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F22 Other classes of groups defined by subgroup chains
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