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Publication . Article . 2001

# p-Divisibility of the Number of Solutions of xp = 1 in a Symmetric Group

Hideki Ishihara; Hiroyuki Ochiai; Yugen Takegahara; Tomoyuki Yoshida;
Closed Access

Published: 01 Sep 2001 Journal: Annals of Combinatorics, volume 5, pages 197-210 (issn: 0218-0006, Copyright policy )
Publisher: Springer Science and Business Media LLC
Abstract

For a prime p and for the number a(n) of solutions of xp = 1 in the symmetric group on n letters, ordp$$(a(n)) \geq [n/p] - [n/p^2]$$, and especially, ordp$$(a(n)) = [n/p] - [n/p^2]$$ provided $$n \equiv 0$$ mod p2. Let r be an integer with $$1 \leq r \leq p^2 - 1$$. If ordp$$(a(r)) \leq [r/p] + 1$$, then, for each positive integer m, ordp$$(a(mp^2 + r)) = m(p-1)+ ord_p (a(r))$$. Assume that ordp$$(a(r)) = [r/p] + 2$$. If $$a(p^2 + r) \equiv -p^{p-1} a(r) mod \quad p^{p+[r/p]+2}$$, then ordp$$(a(mp^2 + r)) = m(p-1) + [r/p] + 2$$; otherwise, there exists a p-adic integer b such tha ordp$$(a(mp^2 + r)) = m(p-1) + [r/p] + 2 + ord_p(m-b)$$.

Subjects by Vocabulary

Microsoft Academic Graph classification: Integer Symmetric group Prime (order theory) Mathematics Divisibility rule Discrete mathematics Combinatorics

Subjects

Discrete Mathematics and Combinatorics