We study a new object that can be attached to an abelian variety or complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over field of numbers this is elementary 2-group with explicit upper bound on rank. exhibit many cases in which group zero, and construct varieties every dimension starting 2, both simple non-simple, order 2. also address situation finite characteristic...

A group in which every element commutes with its endomorphic images is called an E-group. If p is a prime number, a p-group G which is an E-group is called a pE-group. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abel...