Cellular covers for R-modules and varieties of groups

Abstract.

Recall the well-known notion of a cellular cover e:HG from algebraic topology (here for groups and R-modules). The map e is a homomorphism such that any homomorphism :HG factors uniquely through e. The existence of cellular covers of groups and R-modules was shown in [14, 31] and a more explicit construction was given in [8, Section 6.3]. We want to base the existence and uniqueness of cellular covers on classical results in group and module theory; see Baer's [2, 3, 4] and Dickson's [13]. With the help of an argument from [8] (partly from [14]) we shift the result from R-modules to the existence of cellular covers over any variety of groups and get the following result. If G belongs to some variety , then any cellular cover 1KCG is a short exact sequence with Cc (and hence K,C,Gc). Here c is the variety generated by all central extensions of groups V/V with V. This unifies older arguments (from [14, 8]) and adds 20 new group varieties. Moreover, the following interesting problem arrives naturally. Can we find group varieties which are not closed under taking cellular covers? The positive answer to this question needs work and application of deep results from combinatorial group theory and on the Schur multiplier; see [29, 34]. We will show that for each prime p>1075 the Burnside-variety p satisfying xp=1 has cellular covers with free abelian subgroups of infinite rank, thus surely these cellular covers of groups from p are outside p.