# On the Sylow subgroup of alternating groups

Let $A_{n}$ be alternating group of degree $n$. It's known if $p$ is the greatest prime not exceeding $n$, then the number of Sylow $p$-subgroup of $A_{n}$ is $n!/p(p-1)(n-p)!$. I would like to know if $r$ is a arbitary prime divisor of order group $A_{n}$, then what is the number of Sylow $r$ -subgroup of $A_{n}$? Also I would like to know whether the number of Sylow $r$-subgroup is a multiple of $p$?

All thoughts appreciated!

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See this article, On the Sylow Subgroups of the Symmetric and Alternating Groups by L. Weisner American Journal of Mathematics Vol. 47, No. 2 (Apr., 1925), pp. 121-124. –  Babak S. Jun 10 '12 at 5:12
@Sara This paper is the best source for your question. –  Babak Miraftab Jun 10 '12 at 5:35
@Babak Sorouh and babgen, thank you very much. –  Sara Jun 10 '12 at 9:34