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| Paper IPM / M / 8585 |
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| Abstract: | |
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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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