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Prove that for any element $b$, $|b|$ divides $|a|$ (order of $b$ divides order of $a$).
Finite abelian group generated by elements of maximal order

Let $G$ be a finite abelian group. Let $a\in G$ be an element of maximal order. Prove $|b|$ divides $|a|$ for all $b\in G$

I have been given a general outline of a proof but I am stuck on a few parts: I suppose that $g$ is an element of maximal order in $G$ and $h$ is a non identity element of $G$.

I am stuck at a few parts namely,

showing $|gh|\gt |g|$ and from there assuming $|h|$ doesn't divide $|g|$ proving that there exist integers i,j such that $ |g^i h^j |> |g|$

Any help would be appreciated,


This was the basic outline (Note that if g and h are elements of a group and $gh=hg$, it is possible to show that $(gh)^n = g^n h^n$ for any integer n. If you want to use this fact, please include a short proof.)

Suppose that g is an element of maximal order. Let h be a nonidentity element of G.

Step 1: Assume that $gcd(|g|,|h|)=1$. Prove that $|gh| > |g|$.

Step 2: Assume that $|h|$ does not divide $|g|$. Prove that there exist integers i and j such that $|g^i h^j |> |g|$.

  • 1
    $\begingroup$ These steps sound, to me, very specific to the outline we cannot see. For instance, it is impossible to show $|gh|>|g|$ in general, so that means there is some assumption in the outline that we do not know. I think it is best just to copy the outline into your question. $\endgroup$ Jan 30, 2013 at 0:58
  • $\begingroup$ @peoplepower hopefully that helps, this was the outline i was given. $\endgroup$
    – bobdylan
    Jan 30, 2013 at 3:10
  • $\begingroup$ I take it you want this question to be reopened because you feel it differs significantly from the other one. You can ask for this by posting an answer to meta.math.stackexchange.com/questions/6424/… being sure to explain why you believe your question is not a duplicate. $\endgroup$ Jan 30, 2013 at 3:35


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