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Two functions from Abelian groups to perfect groups. (English) Zbl 0614.20038
Given a perfect group P, it is well-known that one can construct the universal central extension $$\hat P$$ of P; there is a universal exact sequence $$1\to H\to \hat P\to P\to 1$$. Then $$H=Z(\hat P)$$ and $$\hat P$$ itself has trivial universal central extension; H is the Schur multiplier $$H_ 2P.$$
In this note, the author reverses this construction. Given an abelian group G, he constructs (functorially) a group A(G) which is acyclic $$(H_ iA(G)=0$$ all $$i>0)$$ and has a natural isomorphism Z(A(G))$$\cong G$$; if $$B(G)=A(G)/G$$, then $$1\to G\to A(G)\to B(G)\to 1$$ is the universal central extension of B(G).
The group A(G) is a variant of an upper triangular matrix ring over the integers. Choose a dense linear ordered set $$\Lambda$$ with initial and final elements. Then A(G) has generators $$X^ a_{\lambda \mu}$$, $$\lambda <\mu$$, where $$a\in {\mathbb{Z}}$$ for $$0\neq \lambda$$ and $$a\in G$$ if $$0=\lambda$$, and relations $$X^ a_{\lambda \mu}X^ b_{\lambda \mu}=X_{\lambda \mu}^{a+b}$$, $$[X^ a_{\lambda \mu},X^ b_{\delta \eta}]=1$$ if $$\mu\neq \delta$$, $$1\neq \eta$$, and $$[X^ a_{\lambda \mu},X^ b_{\delta \eta}]=X^{ab}_{\lambda \eta}$$ if $$\mu =\delta$$ regarding $${\mathbb{Z}}$$ as trivial G-module.
Reviewer: M.E.Keating

##### MSC:
 20J05 Homological methods in group theory 20F05 Generators, relations, and presentations of groups 20E22 Extensions, wreath products, and other compositions of groups 55P20 Eilenberg-Mac Lane spaces
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