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On square roots of the uniform distribution on compact groups

Authors: Persi Diaconis and Mehrdad Shahshahani
Journal: Proc. Amer. Math. Soc. 98 (1986), 341-348
MSC: Primary 22C05; Secondary 43A05
MathSciNet review: 854044
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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.

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Keywords: Compact groups, factorization, Haar measure, normality of closed subgroups
Article copyright: © Copyright 1986 American Mathematical Society