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Let $G$ be a cyclic subgroup of order $n$, generated by say $a\in G$ where the identity of $G$ is labelled $e$. Let $H$ be the cyclic subgroup of $G$ generated by some $a^{m}\in G$. Then I want to show that the order of $H$ is equal to $n/d$ where $d$ is the greatest common divisor of $n$ and $m$.

So far I've got:

This can be reduced to the statement: If $b = a^{m}$, the smallest positive integer $k$ such that $b^{k} = e$ is $n/d$, where $d$ is the greatest common divisor of $n$ and $m$.

Step 1: I've shown that $b^{n/d} = e$ (the easy part).

But this step is giving me problems.

Step 2: Show that there is no $k < n/d$ such that $b^{k} = e$.

I'm reading this from the book "A First Course in Abstract Algebra" by John B. Fraleigh, but I cannot follow his proof. I can supply his argument if anyone requests it.

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You haven't said what $b$ is - presumably you intended $b=a^m$. Then you need to replace $a$ with $b$ in your statements of both Step 1 and Step 2. –  Zev Chonoles Jun 1 '12 at 20:44
I have edited the question to reflect your corrections. Sorry that was quite a blunder... :P –  roo Jun 1 '12 at 21:58
I'm reading Theorem 6.14 of Fraleigh's book right now, which I assume you're referencing. What part of his proof do you find confusing? Maybe someone here can clarify it. –  yunone Jun 1 '12 at 22:09
I'm actually reading an earlier edition to the one you are referring to. I've actually figured it out, and I'll answer my own question in a moment. Thanks very much for checking though! –  roo Jun 1 '12 at 22:26

3 Answers 3

up vote 3 down vote accepted

Let $b=a^m$, and let $d=\gcd(m,n)$, $m'=\frac{m}d$, and $n'=\frac{n}d$. Suppose that $b^k=e$, where $k>0$. Then $a^{km}=e$, so $km$ is a multiple of $n$; say $km=rn$ for some $r>0$. Divide through by $d$: $km'=rn'$, so $km'$ is a multiple of $n'$, where $m'$ and $n'$ are relatively prime. Can you finish it from here?

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@Dylan: I sure do; thanks. –  Brian M. Scott Jun 1 '12 at 22:52

I ended up relying on a lemma from number theory: If $a$ divides $bc$, then $\frac{a}{d}$ divides $c$, where $d = gcd(a,b)$ (a tedious but simple verification using Fundamental Theorem of Arithmetic).

Then if $(a^{s})^{k} = e$, then $n|sk$. Therefore by the lemma, $\frac{n}{d} | k$, where $d = gcd(n,s)$. This is the remaining step to prove in my OP.

Thanks for your attention all. Sorry if I wasted anyone's time!

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If $a$ is the generator of $G$ and $|G|=n$, then $a^{n/d}\neq e$. You want to find the smallest $k$ such that $(a^m)^k=e$, that is, what is $|a^m|$?

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This is what I meant to say, sorry for the error. –  roo Jun 1 '12 at 21:59

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