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Let G be a commutative group, and a, b ∈ G. Suppose that ord(a) = m, ord(b) = n, and gcd(man) = 1. Prove that there exists an element in G or order mn.

What I've done so far, is I tried to find the order of ab. I can prove that the order is divisible by lcm(m,n)gcd(m,n) but I can't prove whether it is equal to lcm(m,n)...

Sugar Bradshaw
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1 Answers1

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Suppose $(ab)^k=1$. Then $x=a^k=b^{-k}$ belongs to $\langle a\rangle\cap\langle b\rangle$.

Prove that $\langle a\rangle\cap\langle b\rangle=\{1\}$ (use Lagrange).

Then $x=1$, so $a^k=1$ and $b^k=1$. Therefore $m\mid k$ and $n\mid k$.

Prove that $mn\mid k$ (use $\gcd(m,n)=1$).

On the other hand, $(ab)^{mn}=1$. Conclude.

egreg
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