2-adic properties of the numbers of representations in wreath products.

Let C 2 ≀ H n denote the wreath product of a cyclic group C 2 of order 2 with a subgroup H n of the symmetric group S n on n letters, and let A be the direct product of a cyclic group of order 2 u with u ≥ 1 and a cyclic group of order 2 v with u ≥ v ≥ 0 . The number of homomorphisms from A to C 2 ≀ H n is denoted by h (A , C 2 ≀ H n) . Let A n be the alternating group on n letters. When H n = A n , h (A , C 2 ≀ A n) has some 2-adic properties which are similar to those of h (A , C 2 ≀ S n) . The exponent of 2 in the decomposition of h (A , C 2 ≀ H n) into prime factors is denoted by ord₂ (h (A , C 2 ≀ H n)) . Let [ x ] denote the largest integer not exceeding a real number x . For each n , τ ¯ 2 (u , v) (n) denotes ∑ j = 0 u - 1 [ n / 2 j ] + [ n / 2 u + 1 ] - [ n / 2 u + 2 ] if u = v and denotes ∑ j = 0 u - 1 [ n / 2 j ] - (u - v) [ n / 2 u ] if u ≥ v + 1 , which is known to be the lower bound of ord₂ (h (A , C 2 ≀ S n)) . There are three types of 2-adic properties of h (A , C 2 ≀ A n) . The lower bound of ord₂ (h (A , C 2 ≀ A n)) is τ ¯ 2 (u , v) (n) if u + δ v 0 ≤ v + 2 , and is τ ¯ 2 (u , v) (n) - 1 if u + δ v 0 ≥ v + 3 . Suppose that u + δ v 0 ≥ v + 2 . For any positive odd integer y , ord₂ (h (A , C 2 ≀ A 2 u y)) and ord₂ (h (A , C 2 ≀ A 2 u y + 1)) are described by certain 2-adic integers. The values { h (A , C 2 ≀ A n) } n = 0 ∞ are explained by certain 2-adic analytic functions. The results are obtained by using an explicit description of the generating function ∑ n = 0 ∞ h (A , C 2 ≀ A n) X n / (2 n n !) . [ABSTRACT FROM AUTHOR]