On Sidki's presentation for orthogonal groups

Justin McInroy*, Sergey Shpectorov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
7 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)227-248
Number of pages22
JournalJournal of Algebra
Early online date17 Apr 2015
Publication statusPublished - 5 Jul 2015


  • Clifford algebra
  • Matrix groups over rings
  • Orthogonal groups
  • Presentations

ASJC Scopus subject areas

  • Algebra and Number Theory


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