Prove in a group, if gcd(|a|,|b|) = 1, then ⟨a⟩ ∩ ⟨b⟩ = {e} (ILIEKMATHPHYSICS)

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

Prove if gcd(a,b) = 1 then gcd(a^n,b^n) = 1 [ILIEKMATHPHYSICS]

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If gcd(a,b) = 1 then gcd(a^2,b^2) = 1 (Proof) [ILIEKMATHPHYSICS]

Proof: Group Element is the Inverse of its Inverse | Abstract Algebra