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

Abstract

We study the 2-adic properties for the numbers of involutions in the alternative groups, and give an affirmative answer to a conjecture of Kim and Kim [A combinatorial approach to the power of 2 in the number of involutions, J. Combin. Theory Ser. A117 (2010) 1082–1094]. Some analogous and general results are also presented.

On the number of even roots of permutations

“…This generating function was used in the work of Koda, Sato and Tskegahara [9]. For the case of odd roots…”

p-adic estimates of the number of permutation representations

“…Hence the assertion is a consequence of Lemma 2.6. □ Remark 3.12 The assertions of Theorems 3.6, 3.9, and 3.11 with v = 0 are given in [12].…”

“…Using E p (X) and E C p s (X), Conrad [2] presented some p-adic properties of h n (C p s ). We attempt to adapt his methods for the study of h n (G) on the basis of some facts given in [12,20].…”

“…Let h n (A; C p ) be the number of homomorphisms from A to C p ≀ S n , and set h 0 (A; C p ) = 1. By [14, Corollary 1], When v = 0, the p-adic properties of h n (P ; C p ) are explored in [12]. For example,…”