Engel-like Identities Characterizing Finite Solvable Groups
| dc.creator | Bandman, Tatiana | |
| dc.creator | Greuel, Gert-Martin | |
| dc.creator | Grunewald, Fritz | |
| dc.creator | Kunyavskii, Boris | |
| dc.creator | Pfister, Gerhard | |
| dc.creator | Plotkin, Eugene | |
| dc.date | 2003-03-13 | |
| dc.date.accessioned | 2026-07-07T04:56:01Z | |
| dc.date.available | 2026-07-07T04:56:01Z | |
| dc.description | 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 $u_k\equiv 1$ in a group $G$ implies $u_m\equiv 1$ 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 $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. 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) . | |
| dc.description | 63 pages, LateX | |
| dc.identifier | https://arxiv.org/abs/math/0303165 | |
| dc.identifier | http://arxiv.org/abs/math/0303165 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66782 | |
| dc.subject | Group Theory | |
| dc.subject | 20F16 (Primary) 20E34,14Gxx,14-04 (Secondary) | |
| dc.title | Engel-like Identities Characterizing Finite Solvable Groups | |
| dc.type | text |