Research on p-adic properties of the numbers of permutation representations

有三个结果。1)得到了有限交换p-群到对称群的同态个数的p-进的性质。特别地，明确了这类数分解为素因数的p的指数。有6种不同类型的结果。2)1的指数加上2阶循环群的交替群或圈积中的对合的个数被交错群分解为素因数时，2的指数可描述为2进整数。3)得到了从两个循环P-群的直积到一个循环P-群的圈积的对称群的同态个数的p-进的性质。这些结果与对称群的同态个数的结果密切相关。

There are three results. 1) p-adic properties of the number of homomorphisms from a finite abelian p-group to symmetric groups were obtained. In particular, the exponent of p of the decomposition of such numbers into prime factors was made clear. There are 6 different types of the results. 2) The exponent of 2 in the decomposition of one plus the number of involutions in alternating groups or wreath products of a cyclic group of order 2 by alternating groups groups into prime factors could be described as 2-adic integers. 3)p-adic properties of the number of homomorphisms from the direct product of two cyclic p-groups to wreath product of a cyclic p-group by symmetric groups were obtained. The results are closely related to that for the number of homomorphisms to symmetric groups.