Automorphisms of extra special groups and nonvanishing degree 2 cohomology

Robert L. Griess, Jr.

Vol. 48 (1973), No. 2, 403–422

DOI: 10.2140/pjm.1973.48.403

Abstract

If E is an extra-specia12-group, it is known that Aut (E)∕ Inn (E) is isomorphic to an orthogonal group. We prove that this extension is nonsplit, except in small cases. As a consequence, the nonvanishing of the second cohomology groups of certain classical groups (defined over F₂) on their standard modules may be inferred. Also, a criterion for a subgroup of these orthogonal groups to have a nonsplit extension over the standard module is given.

Mathematical Subject Classification 2000

Primary: 20J05

Secondary: 10C30

Milestones

Received: 25 May 1972

Revised: 8 March 1973

Published: 1 October 1973