Cauchy's Group Theorem/Proof 1

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Theorem

Let $G$ be a finite group whose identity is $e$.

Let $p$ be a prime number which divides order of $G$.


Then $G$ has a subgroup of order $p$.


Proof

Let $\order G$ be a prime number.

Then from Prime Group is Cyclic $G$ has a generator $\gen g$ such that $\order g = p$.


Now suppose $e \ne g \in G$.

Let $\order g = n$.

Let $p \divides n$.

Then by Subgroup of Finite Cyclic Group is Determined by Order the cyclic group $\gen g$ has an element $g$ of order $p$.


Suppose $p \nmid n$.

From Subgroup of Abelian Group is Normal, $\gen g$ is normal in $G$.

Consider the quotient group $G' = \dfrac G {\gen g}$.

As $p \nmid n$, $\gen e \subsetneq \gen g \subsetneq G$.

Thus $\order {G'} < \order G$.

But we have that $p \divides \order G$.


It follows by induction that $G'$ has an element $h'$ of order $p$.

Let $h$ be a preimage of $h'$ under the quotient epimorphism $\phi: G \to G'$.

Then:

$\paren {h'}^p = e'$

where $e'$ is the identity of $G'$.

Thus $h^p \in \gen g$ so $\paren {h^p}^n = \paren {h^n}^p = e$.

Thus either $h^n$ has order $p$ or $h^n = e$.

If $h^n = e$ then $\paren {h'}^n = e'$.

But since $p$ is the order of $h'$ it would follow that $p \divides n$, contrary to assumption.

$\blacksquare$


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