# $Q$-perfect groups and universal Q-central extensions

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- Publicacions Matemàtiques
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## Abstract

Publicacions Matemátiques, Vol 34 (1990), 291-297 . Abstract Q-PERFECT GROUPS AND UNIVERSAL Q-CENTRAL EXTENSIONS Using results of Ellis-Rodríguez Fernández, an explicit description by gen- erators and relations is given of the mod q Schur multiplier, and this is shown to be the kernel of a universal q-central extension in the case of a q-perfect group, i .e . one which is generated by commutators and q-th powers . These results generalise earlier work K . Dennis and Brown-Loday. A group G will be called q-perfect, where q is a non-negative integer, if G is generated by its commutator subgroup [G, G] and the elements of the form gq for all g E G . An extension of groups 1 ---L A ---, E ---+ G ---L 1 will be called q-central if it is a central extension and every element of A has order dividing q . The q-central extension (1) will be called universal if for any other q-central extensión 1->A'->E'-->G-+ 1 there is a unique morphism of extensions RONALD BROWN The existence of universal q-central extensions of q-perfect groups in the clas~ical case q = 0 is well known, and a similar argument yields the general case . That is, the universal coefficient theorem yields an exact sequence 0 ---> Ext(Hi(G), H2(G, Zq)) -+ H2 (G, H2(G, Zq)) -> Hom(H2(G), H2(G,71q )) -> 0. 1 A E G 1 1 --> A' > E' i > G -> 1 292 R. BROWN The q-perfect condition implies that the Ext term is zero, and so there is a unique element e in H2 (G, H2(G, Zq)) which maps to the canonical morphism H2(G) -> H2(G, Zq ) induced by the coefficient morphism Z -+ Zq. This coho- mology element e determines the universal q-central extension . However in the case q = 0 Miller in [M] gives for this extension group an explicit construction by generators and relations (see [K] for a recent account, and [B-L] for an account deduced from the non-abelian tensor product) . The aim of this paper is to show that results of [E-R] yield a similar construction, in terms of generators and relations, for the group E

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