The Sylow subgroups of $A_5$

Given: The hypotheses and objects in the Example.

For a finite group $G$ and a prime $p$, let ${\mathrm{Syl}}_p(G)$ be the set of Sylow $p$-subgroups (def-sylow-p-subgroup). Define $n_p(G):=\mathrm{\mid }{\mathrm{Syl}}_p(G)\mathrm{\mid }\mathrm{.}$ This cardinal is defined even before existence is proved because ${\mathrm{Syl}}_p(G)$ is a subset of the finite power set of $G$; thm-sylow-first-theorem later shows it is nonzero. (The number $n_p(G)$ of Sylow $p$-subgroups).

Let $\mathrm{\mid }G\mathrm{\mid }=p^{a}m$ with $p\nmid m$. Then the number of Sylow $p$-subgroups satisfies $n_p(G)\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p),n_p(G)\mid m\mathrm{.}$. (Sylow III: $n_p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p)$ and $n_p\mid m$ when $\mathrm{\mid }G\mathrm{\mid }=p^{a}m$ with $p\nmid m$).

For $n\in \mathbb{N}$, the alternating group is the kernel of the sign homomorphism, $A_n:=\ker (\mathrm{sgn}:S_n\to \{+1,-1\})=\{\sigma \in S_n:\mathrm{sgn}(\sigma )=1\}\mathrm{.}$ Thus $A_n$ consists exactly of the even permutations. The subgroup and normality assertions implicit in the word “group” follow from thm-image-subgroup-and-kernel-normal. (The alternating group $A_n=\ker (\mathrm{sgn})$ of even permutations).

The conjugacy classes of $S_n$ are in bijection with the tuples of nonnegative integers $(c_1,\dots ,c_n)$ satisfying ${\sum }_{k=1}^{n}k{c}_k=n\mathrm{.}$ For $n=0$, the unique empty tuple indexes the identity class of $S_0$. (The conjugacy classes of $S_n$ are indexed by the tuples $(c_1,\dots ,c_n)$ with $\sum k{c}_k=n$).