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Engel-like Identities Characterizing Finite Solvable Groups | | Tatiana Bandman
; Gert-Martin Greuel
; Fritz Grunewald
; Boris Kunyavskii
; Gerhard Pfister
; Eugene Plotkin
; | | Date: |
13 Mar 2003 | | Subject: | Group Theory MSC-class: 20F16 (Primary) 20E34,14Gxx,14-04 (Secondary) | math.GR | | Abstract: | 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 $u_1,...,u_n,... $ is called correct if $u_kequiv 1$ in a group $G$ implies $u_mequiv 1$ in a group $G$ for all $m>k$. We are looking for an explicit correct sequence of words $u_1(x,y),...,u_n(x,y),...$ such that a group $G$ is solvable if and only if for some $n$ the word $u_n$ is an identity in $G$. Let $u_1=x^{-2}ymin x$, and $u_{n+1} = [xu_nxmin,yu_nymin]$. The main result states that a finite group $G$ is solvable if and only if for some $n$ the identity $u_n(x,y)equiv 1$ 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) . | | Source: | arXiv, math.GR/0303165 | | Services: | Forum | Review | PDF | Favorites | |
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