A Group-Theoretic Consequence of the Donald-Flanigan Conjecture

Abstract

AbstractThe Donald-Flanigan conjecture asserts that for any finite group G and prime p dividing its order #G, the group algebra FpG can be deformed into a semisimple, and hence rigid, algebra. We show that this implies that there is some element g ∈ G whose centralizer CG(g) has a normal subgroup of index p. The method is to observe that the Donald-Flanigan deformation must be a jump, whence, from the deformation theory, H1(FpG, FpG) ≠ 0. Using a standard result linking Hochschild and group cohomology one sees that some H1(CG(g), Fp) must be non-zero, giving the result. (Our corollary to the D-F conjecture has recently been verified by Fleischmann, Janiszczak, and Lempken using the classification of finite simple groups.