Affine Geometry and Engel-like identities for Finite Solvable Groups
by
Tatiana Bandman
Dept. of Mathematics, Bar-Ilan University,Israel
Coauthors: G.-M. Greuel (University of Kaiserslautern), F. Grunewald (Heinrich Heine University), B. Kunyavskii (Bar-Ilan University), G. Pfister (University of Kaiserslautern), Eu. Plotkin (Bar-Ilan University).

A subject of the communication is the Affine Geometry aspect of a joint work of T. Bandman, G.-M. Greuel, F. Grunewald, B. Kunyavskii , G. Pfister, and Eu. Plotkin. We characterize solvable groups in the class of finite groups by identities in two variables.

We define a sequence: u₁(x, y):=x^-2y^-1x, and inductively u_n+1(x, y):=[ x u_n (x, y) x^-1,  y u_n(x, y) y^-1 ]. Our main result is

Theorem 1. A finite group G is solvable if and only if for some n the identity u_n(x, y) = 1 holds in G.

Although the Theorem is a purely group-theoretic result, its proof involves surprisingly diverse methods of algebraic topology, algebraic geometry, arithmetic geometry, group theory, and computer algebra.

The "only if" part of the Theorem is trivial. The non-trivial direction of the Theorem follows immediately from the following

Theorem 2. Let G be a finite non-abelian simple group. Then there are elements x, y of G such that u₁(x, y) ¹ 1 and u₁(x, y)=u₂(x, y).

Using Thompson's list of the minimal simple non-solvable groups we only need to prove Theorem 2 for the groups G in the following list.

(1) G=PSL (3, F₃),

(2) G=PSL (2, F_q) where q > 3 (q=pⁿ, p a prime),       

(3) G=Sz (2ⁿ), n > 2 and odd.

Here F_q stands for the finite field with q elements and Sz (2ⁿ) ( n > 2) denote the Suzuki groups.

For small groups from this list it is a computer task to verify Theorem 2. There are for example altogether 44928 suitable pairs x, y in the group PSL (3, F₃).

The general idea of our proof can be roughly described as follows. For a group G in the above list, using a matrix representation over F _q we interpret solutions of the equation u₁(x, y)=u₂(x, y) as F_q-rational points of an affine variety V_G.

In case (2) this variety does not depend on n. We investigate geometry and topology of this variety in order to use Hasse-Weil type estimates for the number of rational points on a variety defined over a finite field, which guarantee the existence of such points for big q.

In case (3) variety V_G for Suzuki group G depends on n. We manage to find a closed affine absolutely irreducible affine subset V of A⁸ together with the endomorphism a of A⁸ such that

(1) V is invariant under a,

(2) a² is the Frobenius map.

(3) the elements of the group G=Sz (2ⁿ) are precisely the points of V which are fixed for aⁿ. In order to use the Lefschetz trace formula and to show that such points exist, we had to analyze this affine set, i.e to find its singular locus, to estimate betti numbers and to understand what happens at infinity.

Date received: December 22, 2004