Engel-like Identities Characterizing Finite Solvable Groups

Details Engel-like Identities Characterizing Finite Solvable Groups Engel-like Identities Characterizing Finite Solvable Groups

2003-03-13

arxiv.org/abs/math/0303165v1

Mathematics

Group Theory

63 pages, LateX

Scientific paper

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) .

Affiliated with

Also associated with

No associations

Rating

Engel-like Identities Characterizing Finite Solvable Groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Engel-like Identities Characterizing Finite Solvable Groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Engel-like Identities Characterizing Finite Solvable Groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-562290