1Dipartimento di Matematica e Informatica Università di Udine Via delle Scienze 208 I-33100 Udine, Italy 2Department 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
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 $G$.
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
Affiliations des auteurs :
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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author = {M. S. Lucido and M. R. Pournaki},
title = {Probability that an element of a finite group has a
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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
