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**Patriotx12** Prove by induction:

1. If g and h are commuting elements of G, prove g commutes with every positive integer power of h: gh^{n}= h^{n}g for all positive integers n.Then prove g^{m}h^{n }= h^{n}g^{m} for all positive integers m and n. Is this true when m and n are arbitrary integers?

2. If g and h are commuting elements of G, prove (gh)^{n} = g^{n}h^{n} for all positive integers n.

Here's my attempt at solving 1, I'm just not sure if it's correct, or if more needs to be proved.

gh^{n} = h^{n}g

Base step: let n = 1

gh = hg