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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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