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

Abstract

For a prime p and for the number a(n) of solutions of x p = 1 in the symmetric group on n letters, ord p (a(n)) ≥ [n/p] − [n/p 2 ], and especially, ord p (a(n)) = [n/p] − [n/p 2 ] provided n ≡ 0 mod p 2 . Let r be an integer with 1 ≤ r ≤ p 2 − 1. If ord p (a(r)) ≤ [r/p] + 1, then, for each positive integer m, ord p (a(mp 2 + r)) = m(p − 1) + ord p (a(r)). Assume that ord p (a(r)) = [r/p] + 2. If a(p 2 + r) ≡ −p p−1 a(r) mod p p+[r/p]+2 , then ord p (a(mp 2 + r)) = m(p − 1) + [r/p] + 2; otherwise, there exists a p-adic integer b such that ord p (a(mp 2 + r)) = m(p − 1) + [r/p] + 2 + ord p (m − b).

On some properties of the number of permutations being products of pairwise disjoint d-cycles

“…We give a new proof of this known fact in order to obtain some results on periodicity of sequences Hp(n) p βn (mod p r ) n∈N , where r ∈ N + and β n is mentioned lower bound of ν p (H p (n)) or β n = ν p (H p (n)). Moreover, we show that for each prime number p and j ∈ {0, ..., p − 1} there exists a b j ∈ {0, ..., p − 1} such that H p (kp 2 + jp + b j ) = k(p − 1) + j for any k ∈ N. These results can be seen as a complement of the series of papers [5,10,6,7]. In Section 5 we describe the p-adic valuations of the numbers H d (n), where p > d. In order to do this, we show that the sequence (H d (n)) n∈N is a restriction of a differentiable function f d : Z p → Z p , where Z p is a ring of p-adic integers.…”

“…In particular, they obtained several combinatorial identities, presented description of the 2-adic valuation of H 2 (n) and gave precise information about the rates of growth of H 2 (n). The mentioned paper can be seen as a complement of case p = 2 of the papers [5,10,6,7] concerning p-adic valuations of numbers H p (n), where p is a prime number. The study of the sequence (H p (n)) n∈N with p-prime, is quite natural because the number H p (n) counts the elements of order p in the group S n .…”

2-adic properties for the numbers of involutions in the alternating groups

“…The results are divided into three theorems, which generalize part of the results proved by K. Conrad [4] (see also [11,16]…”

“…Corollary 4.5) was given in [6,7,9], which was also shown by various methods (cf. [4,11,13,14]); moreover, the equality holds for all n such that n − [n/p 2 ]p 2 ≤ p − 1 (see, e.g., [6,11,13]). When p = 2, this formula was found by S. Chowla, I.…”

A combinatorial approach to the power of 2 in the number of involutions

“…Example 2.1. Let π ∈ S 29,3 be the following permutation in cycle notation: Then f 3 (π ) = {(1, 2, 3) 3 , (4, 5, 6), (4), (4,6,5), (5), (6), (7,7,7), (8) 3 , (9) 2 , (9, 10, 10)}. The permutation π belongs to an equivalence class (7,7,7); (1, 2, 3) 3 , (4, 5, 6), (4, 6, 5), (9, 10, 10); (4), (5), (6), (9) (4,5,6), (4, 6, 5), (9, 10, 10), (4), (5), (6), (9) 2 .…”