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

Abstract

Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup $G^{\prime }$ G ′ . We prove that the exponent and the abelianization of the centralizer of $G^{\prime }$ 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 $G^{\prime }$ G ′ is determined. These claims are known to be false for p = 2.

Non-isomorphic 2-groups with isomorphic modular group algebras

“…These groups were discovered during an effort to solve the modular isomorphism problem for two-generated finite p-groups with cyclic derived subgroup, after these had been classified by Broche, García-Lucas and del Río [4]. Although considerable effort has been devoted to solving the problem in this wide class, it has not been achieved completely so far [9,8]. We consider the negative solutions from another viewpoint and embed them in a natural class of groups for which we can completely solve the modular isomorphism problem.…”