2003/03/13 by Tatiana Bandman, Gert-Martin Greuel, Bandman, Tatiana +9
Mathematics · #14-04 (Secondary) #14Gxx #20F16 (Primary) 20E34 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:14-04 #msc:14Gxx #msc:20E34 #msc:20F16
paper · pdf · doi:10.48550/arxiv.math/0303165
63 pages, LateX
arxiv created 2003/03/13 · arxiv updated 2009/11/30
In the paper we characterize the class of finite solvable groups by two-variable identities in a way similar to the characterization of finite nilpotent groups by Engel identities. More precisely, a sequence of words u1,...,un,... is called correct if uk≡ 1 in a group G implies um≡ 1 in a group G for all m>k. We are looking for an explicit correct sequence of words u1(x,y),...,un(x,y),... such that a group G is solvable if and only if for some n the word un is an identity in G. Let u1=x-2ymin x, and un+1 = [xunxmin,yunymin]. The main result states that a finite group G is solvable if and only if for some n the identity un(x,y)≡ 1 holds in G. In the language of profinite groups this result implies that the provariety of prosolvable groups is determined by a single explicit proidentity in two variables. The proof of the main theorem relies on reduction to J.Thompson's list of minimal non-solvable simple groups, on extensive use of arithmetic geometry (Lang - Weil bounds, Deligne's machinery, estimates of Betti numbers, etc.) and on computer algebra and geometry (SINGULAR, MAGMA) .