群论/单群与西罗定理

定义（西罗 p-子群）:

令 $G$ 是一个群，令 $p\in \mathbb {N}$ 是一个素数，使得 $p||G|$。那么 $G$ 的一个西罗 $p$-子群 是一个子群 $P\leq G$，使得 $|P|=p^{k}$，其中 $k$ 是使 $p^{k}||G|$ 成立的最大值。

定理（柯西定理）:

令 $G$ 是一个群，其阶 $|G|$ 可被素数 $p\in \mathbb {N}$ 整除。那么 $G$ 包含一个阶为 $p$ 的元素。

证明： $G$ 通过共轭作用于自身。令 $x_{1},\ldots ,x_{n}\in G$ 是共轭类的代表系。类方程给出

$|G|=\sum _{k=1}^{n}[G:G_{x_{k}}]$.

Either, there exists $k\in \{1,\ldots ,n\}$ such that $[G:G_{x_{k}}]$ is both not $1$ and not divisible by $p$, in which case we may conclude by induction on the group order, noting that $p$ divides $|G_{x_{k}}|$ and $|G_{x_{k}}|<|G|$, or for all $k\in \{1,\ldots ,n\}$ the number $[G:G_{x_{k}}]$ is either $1$ or divisible by $p$; but in this case, by taking the class equation $\mod p$, we obtain that $Z(G)$ is nontrivial and moreover that its order is divisible by $p$. Hence, it suffices to consider the case where $G$ is an abelian group. Take then any element $h\in G$. If $h$ has order divisible by $p$, raising $h$ to a sufficiently high power will produce an element of order $p$. Otherwise, the order of $G/\langle h\rangle$ is divisible by $p$, and by induction we find an element $g+\langle h\rangle \in G/\langle h\rangle$ whose order is divisible by $p$. Then the order of $g$ will also be divisible by $p$, because otherwise, passing to the quotient, $(g+\langle h\rangle )^{k}=e+\langle h\rangle$ for some $k$ not divisible by $p$. $\Box$

定理（西罗定理）:

令 $G$ 是一个有限群，使得 $|G|=p^{k}m$，其中 $p\nmid m$。那么以下成立

1. $G$ 具有一个 Sylow 子群。
2. $G$ 对 Sylow 子群的共轭作用是可迁的。
3. 如果 $n_{p}$ 是 Sylow $p$-子群的数量，则 $n_{p}|m$ 且 $n_{p}\equiv 1\mod p$。
4. 每个 $p$-子群 $G$ 都包含在某些 Sylow $p$-子群 $G$ 中。

定义 (简单群):

如果 $G$ 和 $\{e\}$ 是 $G$ 唯一的正规子群（其中 $e\in G$ 表示单位元），则群 $G$ 是一个简单群。