First Sylow Theorem/Corollary/Proof 1
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Corollary to First Sylow Theorem
Let $p$ be a prime number.
Let $G$ be a group.
Let:
- $p^n \divides \order G$
where:
- $\order G$ denotes the order of $G$
- $n$ is a positive integer.
Then $G$ has at least one subgroup of order $p$.
Proof
Let $\order G = k p^r$ where $p \nmid k$.
From the First Sylow Theorem, $G$ has a subgroup $S$ of order $p^r$.
From (need to find it), $S$ itself has subgroups of order $p^n$ for all $n \in \set {1, 2, \ldots, r}$.
$\blacksquare$
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Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 56$. First Sylow Theorem