2-Adic properties for the numbers of representations in the alternating groups

Abstract

Let A be the direct product of a cyclic group of order $2^u$ with $u\ge 1$ and a cyclic group of order $2^v$ with $u\ge v\ge 0$. There are some 2-adic properties of the number $h(A,A_n)$ of homomorphisms from A to the alternating group $A_n$ on n-letters, which are similar to those of the number of homomorphisms from A to the symmetric group on n-letters. The exponent of 2 in the decomposition of $h(A,A_n)$ into prime factors is denoted by ${\mathrm {ord}}_2(h(A,A_n))$. Let [x] denote the largest integer not exceeding a real number x. For any nonnegative integer n, the lower bound of ${\mathrm {ord}}_2(h(A,A_n))$ is $\sum _{j=1}^u[n/2^j]+[n/2^{u+2}]-[n/2^{u+3}]-1$ if $u=v\ge 1$, and is $\sum _{j=1}^u[n/2^j]-(u-v)[n/2^{u+1}]-1$ otherwise. For any positive odd integer y, ${\mathrm {ord}}_2(h(A,A_{2^{u+1}y}))$ and ${\mathrm {ord}}_2(h(A,A_{2^{u+1}y+1}))$ are described by certain 2-adic integers if either $u\ge v+2\ge 3$ or $u\ge 1$ and $v=0$. The values $\{h(A,A_n)\}_{n=0}^\infty$ are explained by certain 2-adic analytic functions unless $u=v+1\ge 2$. The results are obtained by using the generating function $\sum _{n=0}^\infty h(A,A_n)X^n/n!$.