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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
DOI: https://doi.org/10.1090/S0002-9939-1986-0854044-9
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.


References [Enhancements On Off] (What's this?)

  • [H] Behncke, (1971), Nilpotent elements in group algebras, Bull. Acad. Polon. Ser. Math. 19 (1971), 197-198. MR 0283582 (44:813)
  • [P] Diaconis, (1982), On the use of group representations in probability and statistics, Typed Lecture Notes, Department of Statistics, Institute of Mathematical Statistics, Harvard University (to appear). MR 964069 (90a:60001)
  • [S] Eilenberg and N. Steenrod, (1952), Foundations of algebraic topology, Princeton Univ. Press, Princeton, N.J., 1952. MR 0050886 (14:398b)
  • [M] Hall, (1959), The theory of groups, Macmillan, New York, 1959. MR 0103215 (21:1996)
  • [E] Hewitt and K. Ross, (1963), Abstract harmonic analysis. I, Springer-Verlag, Berlin, 1963.
  • 1. -, (1970), Abstract harmonic analysis. II, Springer-Verlag, Berlin, 1970.
  • [I] Kaplansky, (1952), Infinite abelian groups, Univ. of Michigan Press, Ann Arbor, Michigan, 1952. MR 0065561 (16:444g)
  • [S] Mac Lane, (1975), Homology theory, Springer-Verlag, Berlin, 1975. MR 1344215 (96d:18001)
  • [J] Pascaud, (1973), Anneaux de groups reduits, C. R. Acad. Sci. Paris Ser. A 277 (1973), 719-722. MR 0335570 (49:351)
  • [S] Shatz, (1972), Profinite groups, arithmetic, and geometry, Princeton Univ. Press, Princeton, N.J., 1972. MR 0347778 (50:279)
  • [S] K. Sehgal, (1975), Nilpotent elements in group rings, Manuscripta Math. 15 (1975), 65-80. MR 0364417 (51:671)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0854044-9
Keywords: Compact groups, factorization, Haar measure, normality of closed subgroups
Article copyright: © Copyright 1986 American Mathematical Society

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