Maximal finite order of Abelian Groups

Question

Let $G$ be a commutative group, and let $g$ be an element of $G$ be an element of maximal finite order, then $|h|\leq |g|$. Prove that in fact if $h$ is finite order in $G$, then $|h|$ divides $|g|$.

This is what I have:

Proof by contradiction If $|h|$ is finite but doesn't divide $|g|$, then there is a prime integer $p$ such that $|g|=rp^m$, $|h|=sp^n$, with $r$ and $s$ relatively prime to $p$ and $m < n$.

(This is where I do not know where to go.)

Your conclusion on $r,s,p,m$ is wrong. – Olivier Bégassat Feb 26 '13 at 22:10 — Olivier Bégassat, Feb 26 '13 at 22:10
How? Please explain? – Username Unknown Feb 26 '13 at 22:13 — Username Unknown, Feb 26 '13 at 22:13
Why would $r$ and $s$ be relatively prime to $m$? – Olivier Bégassat Feb 26 '13 at 22:20 — Olivier Bégassat, Feb 26 '13 at 22:20
Sorry there was an error in the question. I fixed it – Username Unknown Feb 26 '13 at 22:23 — Username Unknown, Feb 26 '13 at 22:23

score 3 · Accepted Answer · answered Feb 26 '13 at 22:40

3

Suppose the $|h|$ doesn't divide $|g|$, then $|h|$ contains a prime factor with multiplicity higher than in $|g|$. This tells us that $gh$ has order greater than that of $g$, contradicting the hypothesis.

answered Feb 26 '13 at 22:40

Paul Orland

6,888

That's it!?!?!?!? That was simple. Is this a rigourous proof? – Username Unknown Feb 26 '13 at 22:57
I mean, you might want to flesh out the details, but I'm pretty sure the reasoning is sound. – Paul Orland Feb 26 '13 at 23:11
Since the statement in question is in general false for non-commutative groups, a correct argument needs to use the commutativity. Where did you do so in your answer? – Pete L. Clark Jan 02 '14 at 06:44
Commutativity tells us that $|gh| = \hbox{lcm}(|g|,|h|)$? – Paul Orland Jan 04 '14 at 02:18

Maximal finite order of Abelian Groups

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