On square roots of the uniform distribution on compact groups

Abstract:

Let $G$ be a compact separable topological group. When does there exist a probability $P$ such that $P * P = U$, where $U$ is Haar measure and $P \ne U$? We show that such square roots exist if and only if $G$ is not abelian, nor the product of the quaternions and a product of two element groups. In the course of proving this we classify compact groups with the property that every closed subgroup is normal.