We obtain new bounds for the exponent of the Schur multiplier of a given p-group. We prove that the exponent of the Schur multiplier can be bounded by a function depending only on the exponent of a given group. As a consequence we show that the exponent of the Schur multiplier of any group of exponent four divides eight, and that this bound is best possible. The notion of the exponential rank of a p-group is introduced. We show that powerful p-groups have exponential rank either zero or one.

The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space. We derive a homological version of the Bogomolov multiplier, prove a Hopf-type formula, find a five term exact sequence corresponding to this invariant, and describe the role of the Bogomolov multiplier in the theory of central extensions. A new description of the Bogomolov multiplier of a nilpotent group of class two is obtained. We define the Bogomolov multiplier within $K$-theory and show that proving its triviality is equivalent to solving a long-standing problem posed by Bass. An algorithm for computing the Bogomolov multiplier is developed.

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We prove that the exponent of the nonabelian tensor product of two locally finite groups can be bounded in terms of exponents of given groups. Several estimates for the exponents of nonabelian tensor squares are obtained. In particular, if the group G is nilpotent of class ≤ 3 and of finite exponent, then the exponent of its nonabelian tensor square divides the exponent of G.

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