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 $$\sum _{n=0}^\infty h(A,A_n)X^n/n!$$ .