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.


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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