On Square Roots of the Uniform Distribution on Compact Groups

Authors:
  • PDPersi Diaconis
    Persi Diaconis
  • MSMehrdad Shahshahani
    Mehrdad Shahshahani
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Abstract
Let be a compact separable topological group. When does there exist a probability such that , where is Haar measure and ? We show that such square roots exist if and only if 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.