A group-theoretic consequence of the Donald–Flanigan conjecture

For a finite group G and a prime p dividing the order of G, Donald and Flanigan conjecture that the group algebra FpG can be deformed into a semisimple (hence rigid) algebra. We demonstrate that this implies that for some element g of G, the centralizer CG(g) of g in G has a normal subgroup of index p. The method is to observe that the Donald–Flanigan deformation must be a jump. This implies that there is a non-trivial class in H(FpG, FpG); therefore this Hochschild cohomlogy group must be nontrivial. Using a standard result linking Hochschild and group cohomology one sees that some H(CG(g), Fp) must be non-zero. The result follows immediately.