群論/單群與西羅定理

定義（西羅 p-子群）:

令 $G$ 是一個群，令 $p\in \mathbb {N}$ 是一個素數，使得 $p||G|$。那麼 $G$ 的一個西羅 $p$-子群 是一個子群 $P\leq G$，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 $G$ 是一個群，其階 $|G|$ 可被素數 $p\in \mathbb {N}$ 整除。那麼 $G$ 包含一個階為 $p$ 的元素。

證明： $G$ 透過共軛作用於自身。令 $x_{1},\ldots ,x_{n}\in G$ 是共軛類的代表系。類方程給出

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

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（西羅定理）:

令 $G$ 是一個有限群，使得 $|G|=p^{k}m$，其中 $p\nmid m$。那麼以下成立

1. $G$ 具有一個 Sylow 子群。
2. $G$ 對 Sylow 子群的共軛作用是可遷的。
3. 如果 $n_{p}$ 是 Sylow $p$-子群的數量，則 $n_{p}|m$ 且 $n_{p}\equiv 1\mod p$。
4. 每個 $p$-子群 $G$ 都包含在某些 Sylow $p$-子群 $G$ 中。

定義 (簡單群):

如果 $G$ 和 $\{e\}$ 是 $G$ 唯一的正規子群（其中 $e\in G$ 表示單位元），則群 $G$ 是一個簡單群。