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

中文翻译：

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交替组中表示数的 2-Adic 性质

设 A 为阶为 $$2^u$$ 的循环群与 $$u\ge 1$$ 和阶为 $$2^v$$ 的循环群与 $$u\ge v\ge 0$ 的直积$ . 从 A 到 n 个字母上的交替群 $$A_n$$ 的同态数 $$h(A,A_n)$$ 有一些 2-adic 性质，与来自 A 的同态数的性质相似到 n 个字母上的对称群。将 $$h(A,A_n)$$ 分解为质因数时 2 的指数表示为 $${\mathrm {ord}}_2(h(A,A_n))$$ 。让 [x] 表示不超过实数 x 的最大整数。对于任何非负整数 n，$${\mathrm {ord}}_2(h(A,A_n))$$ 的下限是 $$\sum _{j=1}^u[n/2^j] +[n/2^{u+2}]-[n/2^{u+3}]-1$$ 如果 $$u=v\ge 1$$ ，并且是 $$\sum _{j= 1}^u[n/2^j]-(uv)[n/2^{u+1}]-1$$ 否则。对于任何正奇数 y，$${\mathrm {ord}}_2(h(A, A_{2^{u+1}y}))$$ 和 $${\mathrm {ord}}_2(h(A,A_{2^{u+1}y+1}))$$ 被描述通过某些 2-adic 整数，如果 $$u\ge v+2\ge 3$$ 或 $$u\ge 1$$ 和 $$v=0$$ 。值 $$\{h(A,A_n)\}_{n=0}^\infty $$ 由某些 2-adic 解析函数解释，除非 $$u=v+1\ge 2$$ 。结果是通过使用生成函数 $$\sum _{n=0}^\infty h(A,A_n)X^n/n!$$ 获得的。