Probability that an Element of a Finite Group has an n–th Root A. Sadeghieh 1 , H. Doostie 2 and K. Ahmadidelir 3 1 A. Sadeghieh, Science and Research Branch, Islamic Azad University, P.O. Box:14515/1775, Tehran, Iran, E-maill: sadeghieh@iauyazd.ac.ir 2 H. Doostie, Mathematics Department, Tarbiat Moallem University, 49 Mofateh Ave., Tehran 15614, Iran. E-mail: doostih@saba.tmu.ac.ir, Tel.: +98-21-77507772, FAX: +98-21-77602988 3 K. Ahmadidelir, Science and Research Branch, Islamic Azad University, P.O. Box 14515/1775, Tehran, Iran. E-mail: kdelir@gmail.com, Tel.: +98-411-5405791, FAX:+98-411-3333458 Abstract. The probability that a randomly chosen element in a finite group has a square root, has been investigated by certain authors. In this paper, we generalize this probability to p–th root (p> 2 is a prime number) and give some bounds for it. Also, if we denote this probability by P p (G) for a finite group G, we show that the set {P p (G) | G is a finite group } is a dense subset of the closed interval [0, 1], investigating this bounds for P p (G) promote us to pose an open problem concerning the rational subset of [0, 1]. Keywords: Probability in finite groups, Roots of elements of a finite group. AMS Subject Classification Numbers: 20A05, 20B05, 20C33, 20G40, 20P05. Introduction Let n be a positive integer. An element g of a finite group G is said to have an n–th root if there exists an element h ∈ G such that g = h n . Also, g may have one or more n–th roots, or it may have none. Finally, let G n be the 1