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On Square Roots of the Uniform Distribution on Compact Groups

Persi Diaconis and Mehrdad Shahshahani
Proceedings of the American Mathematical Society
Vol. 98, No. 2 (Oct., 1986), pp. 341-348
DOI: 10.2307/2045709
Stable URL: http://www.jstor.org/stable/2045709
Page Count: 8
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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 ≠ 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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