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

We introduce the β-invariant b(ω) attached to a 4-cohomology class ω∈H4(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 H4(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 H4(G,Z) lies in the nilpotent radical of the cohomology ring (Proposition 4.1); (b) up to gauge equivalence, the twisted quantum double Dω(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).