Geodesic
Probability that an element of a finite group has a square root
M. S. Lucido 1   ; M. R. Pournaki 2
1 Dipartimento di Matematica e Informatica Università di Udine Via delle Scienze 208 I-33100 Udine, Italy
2 Department of Mathematical Sciences Sharif University of Technology P.O. Box 11155-9415 Tehran, Iran and School of Mathematics Institute for Studies in Theoretical Physics and Mathematics P.O. Box 19395-5746 Tehran, Iran
Colloquium Mathematicum, Tome 112 (2008) no. 1, pp. 147-155 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let $G$ be a finite group of even order. We give some bounds for the probability ${\rm p}(G)$ that a randomly chosen element in $G$ has a square root. In particular, we prove that ${\rm p}(G) \leq 1-{\lfloor \sqrt{|G|}\rfloor/|G|}$. Moreover, we show that if the Sylow 2-subgroup of $G$ is not a proper normal elementary abelian subgroup of $G$, then ${\rm p}(G) \le 1-1/\sqrt{|G|}$. Both of these bounds are best possible upper bounds for ${\rm p}(G)$, depending only on the order of $G$.
DOI : 10.4064/cm112-1-7
Keywords: finite group even order bounds probability randomly chosen element has square root particular prove leq lfloor sqrt rfloor moreover sylow subgroup proper normal elementary abelian subgroup sqrt these bounds best possible upper bounds depending only order nbsp

M. S. Lucido  1   ; M. R. Pournaki  2

1 Dipartimento di Matematica e Informatica Università di Udine Via delle Scienze 208 I-33100 Udine, Italy
2 Department of Mathematical Sciences Sharif University of Technology P.O. Box 11155-9415 Tehran, Iran and School of Mathematics Institute for Studies in Theoretical Physics and Mathematics P.O. Box 19395-5746 Tehran, Iran 
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M. S. Lucido; M. R. Pournaki. Probability that an element of a finite group has a
square root. Colloquium Mathematicum, Tome 112 (2008) no. 1, pp. 147-155. doi: 10.4064/cm112-1-7

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