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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.

MSC:

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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