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^' . We prove that the exponent and the abelianization of the centralizer of G^' 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^' is determined. These claims are known to be false for p = 2.