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Paper   IPM / M / 8585
School of Mathematics
  Title:   Probability that an element of a finite group has a square root
  Author(s):  M. R. Pournaki (Joint with M. S. Lucido)
  Status:   Published
  Journal: Colloq. Math.
  Vol.:  112
  Year:  2008
  Pages:   147-155
  Supported by:  IPM
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1−⎣√|G|⎦/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1−1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.

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