Sylow theorem

《代数学引论》 · 第 79 页

以下得到的几条主要定理通常称为西罗定理。

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《Abstract Algebra》 · 第 139 页 · Section 4.5「Sylow's Theorem」，Theorem 18

Theorem 18. (Sylow's Theorem) Let $G$ be a group of order $p^a m$, where $p$ is a prime not dividing $m$. (1) Sylow $p$-subgroups of $G$ exist, i.e., $\operatorname{Syl}_p(G)\neq\varnothing$. (2) If $P$ is a Sylow $p$-subgroup of $G$ and $Q$ is any $p$-subgroup of $G$, then there exists $g\in G$ such that $Q\leq gPg^{-1}$, i.e., $Q$ is contained in some conjugate of $P$. In particular, any two Sylow $p$-subgroups of $G$ are conjugate in $G$. (3) The number of Sylow $p$-subgroups of $G$ is of the form $1+kp$, i.e., $n_p\equiv 1\pmod p$. Further, $n_p$ is the index in $G$ of the normalizer $N_G(P)$ for any Sylow $p$-subgroup $P$, hence $n_p$ divides $m$.