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Proof Order Group Abelian Group Group Finite Superset Implication Prime Mutually Coprime

1	Proof
1	Finite
1	Superset
1	Implication
1	Prime
1	Order Group
1	Abelian Group
1	Group
8	Mutually Coprime

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Order Of Elements In Finite Groups

This article proves two elementary results regarding the orders of group elements in finite groups.

Theorem 1 Let be a finite group, and let $a\in G$ and $b\in G$ be elements of that commute with each other. Let $m=\lvert a\rvert$ , $n=\lvert b\rvert$ . If $\gcd(m,n)=1$ , then $mn=\lvert ab\rvert$ .

Proof. Note first that

$\displaystyle (ab)^{mn}=a^{mn}b^{mn}=(a^m)^n(b^n)^m=e_G$

since

and

commute with each other. Thus $\lvert ab\rvert \leq mn$ . Now suppose $\lvert ab\rvert=k$ . Then

$\displaystyle e_G=(ab)^k=(ab)^{km}=a^{km}b^{km}=b^{km}$

and thus $n\vert km$ . But $\gcd(m,n)=1$ , so $n\vert k$ . Similarly, $m\vert k$ and thus $mn\vert k=\lvert ab\rvert$ . These two results together imply that