We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

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.

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