Prove in a group, if ab = ba and have finite order, then gcd(|a|,|b|) = 1 iff |ab| = |a||b|

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Prove in a group, if ab = ba and have finite order, and ⟨a⟩ ∩ ⟨b⟩ = {e}, then |ab| = lcm(|a|,|b|) Prove in a group, if ab = ba and have finite order, and ⟨a⟩ ∩ ⟨b⟩ = {e}, then |ab| = lcm(|a|,|b|)
a is a group element and a has infinite order,prove that a^m is not equals a^n when m not equals n. a is a group element and a has infinite order,prove that a^m is not equals a^n when m not equals n.
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let g1g2 be two elements of a finite group G, then the order of g1g2 is the same as that of g2g1 let g1g2 be two elements of a finite group G, then the order of g1g2 is the same as that of g2g1