On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic

Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup \(G^{\prime }\). We prove that the exponent and the abelianization of the centralizer of \(G^{\prime }\) in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of \(G^{\prime }\) is determined. These claims are known to be false for p = 2.