Prove by induction:
1. If g and h are commuting elements of G, prove g commutes with every positive integer power of h: ghn= hng for all positive integers n.Then prove gmhn = hngm 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 = gnhn 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.
ghn = hng
Base step: let n = 1
gh = hg
Assume ghk = hkg
ghkh = hkhg
ghk = hkg
Is that done correctly, and if so would it end the proof? And can the others be proved in the same way?