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 -------------------==== Posted via Deja News ====----------------------- http://www.dejanews.com/ Search, Read, Post to Usenet