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The endocenter and its applications to quasigroup representation theory. (English) Zbl 0748.20016

Let \(\mathfrak V\) be a variety of groups, and let \(G\in {\mathfrak V}\), then the endocenter \(Z(G;{\mathfrak V})\) of \(G\) in \(\mathfrak V\) is defined by: \(Z(G;{\mathfrak V})=\bigcup_{G\leq H\in{\mathfrak V}}Z(H)\). Unlike the ordinary centre, the endocenter gives a functor from \(\mathfrak V\) to \(\mathbf{Gp}\), the variety of all groups and it follows that (again unlike the centre) the endocenter is fully invariant. Clearly it will always be a subgroup of the centre. If \({\mathfrak V}=HSP(G)\) and \(Z(G)\) is verbal, then \(Z(G)=Z(G;{\mathfrak V})\). In this paper the concept of the endocenter is used in the study of the universal multiplication groups of groups and quasigroups and several interesting results are given.

MSC:

20E10 Quasivarieties and varieties of groups
20F14 Derived series, central series, and generalizations for groups
20N05 Loops, quasigroups
20E07 Subgroup theorems; subgroup growth
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