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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm112-1-7,
author = {M. S. Lucido and M. R. Pournaki},
title = {Probability that an element of a finite group has a square root},
journal = {Colloquium Mathematicae},
volume = {111},
year = {2008},
pages = {147-155},
zbl = {1146.20019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm112-1-7}
}
M. S. Lucido; M. R. Pournaki. Probability that an element of a finite group has a square root. Colloquium Mathematicae, Tome 111 (2008) pp. 147-155. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm112-1-7/