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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