From: Robin Chapman
Subject: Re: Stem Cover
Date: Tue, 20 Jan 1998 09:35:43 -0600
Newsgroups: sci.math
In article <34C47CF5.167E@univ-tln.fr>,
Philippe LANGEVIN wrote:
>
> What is a << stem cover >> of a group G ?
>
This is taken from Rotman's book on homological algebra. Alas he doesn't
provide proofs.
A central extension of a group G is an exact sequence 1 -> A -> E -> G
-> 1 where A is contained in the centre of E. For a fixed G there's a
category of such things, and if there's an inital object that's called
the universal central extension. The trouble is that (for finite G) a
universal central extension exists iff G is perfect, i.e. G = G' its
commutator subgroup. When G is perfect, the A in the universal central
extension is H_2(G,Z) (second homology with coefficients in the
integers). So moe generally one says that a central extension is a stem
extension if A is contained in E'. A stem cover is then a stem extension
with A equal to H_2(G,Z). These always exist, but may not be unique when
G isn't perfect. Rotman gives the example of G = V_4, the Klein 4-group.
Here the homology group has 2 elements, and both the dihedral and
quaternion groups of 8 elements are stem extensions.
Robin Chapman "256 256 256.
Department of Mathematics O hel, ol rite; 256; whot's
University of Exeter, EX4 4QE, UK 12 tyms 256? Bugird if I no.
rjc@maths.exeter.ac.uk 2 dificult 2 work out."
http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks - Feersum Endjinn
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