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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.
Reviewer: A.Drápal (Praha)

##### 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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