On Sidki's presentation for orthogonal groups

TL;DR

This paper investigates Sidki's presentations of groups conjectured to be finite orthogonal groups in characteristic two, exploring their structure, representations, and connections to known algebraic groups and periodicity patterns.

Contribution

It generalizes Sidki's presentations to an infinite family, relates them to Clifford algebras, and provides explicit representations into orthogonal and pin groups over characteristic two rings.

Findings

Groups $y(m,n)$ are conjectured to be finite orthogonal groups in characteristic two.

Explicit representations of $y(m)$ into orthogonal and pin groups are constructed.

Connections to Sidki's homomorphism and Clifford algebra actions are established.

Abstract

We study presentations, defined by Sidki, resulting in groups $y(m,n)$ that are conjectured to be finite orthogonal groups of dimension $m+1$ in characteristic two. This conjecture, if true, shows an interesting pattern, possibly connected with Bott periodicity. It would also give new presentations for a large family of finite orthogonal groups in characteristic two, with no generator having the same order as the cyclic group of the field. We generalise the presentation to an infinite version $y(m)$ and explicitly relate this to previous work done by Sidki. The original groups $y(m,n)$ can be found as quotients over congruence subgroups of $y(m)$. We give two representations of our group $y(m)$. One into an orthogonal group of dimension $m+1$ and the other, using Clifford algebras, into the corresponding pin group, both defined over a ring in characteristic two. Hence, this gives two different actions of the group. Sidki's homomorphism into $SL_{2^{m-2}}(R)$ is recovered and extended as an action on a submodule of the Clifford algebra.