Reed–Muller codes, the fourth cohomology group of a finite group, and the β-invariant

Abstract

We introduce the β-invariant b ( ω ) attached to a 4-cohomology class ω ∈ H 4 ( G , Z ) , G a finite group. Roughly speaking, b ( ω ) keeps track of the restriction of ω to subgroups of G of order 2. If G is an elementary abelian 2-group, we observe that b defines a natural isomorphism from H 4 ( G , Z ) to the shortened third order Reed–Muller binary code. In general, restricting ω to elementary abelian 2-subgroups produces an array of Reed–Muller codewords which can be exploited. We give two main applications: (a) for many of the larger sporadic simple groups, the 2-part of H 4 ( G , Z ) lies in the nilpotent radical of the cohomology ring (Proposition 4.1); (b) up to gauge equivalence, the twisted quantum double D ω ( G ) has a trivial β-invariant (in the sense of quasi-Hopf algebras) if, and only if, ω is a nilpotent element in the cohomology ring (Proposition 5.2).