Proof check

[Boi]

For the order of an element of a group (and the order of a subgroup) I'll be using the notation $o(g)$ (and $o(H)$ respectively)
For the proof I'll need 2 lemmas (they're pretty obvious results, but they aren't written in the book explicitly, so I've felt the need to prove them)
Lemma 1: if $G$ is a group,

 

[blamocur]

Looks incomplete.

 

[Boi]

Looks incomplete.

Yeah, sorry, I misclicked.
I'll make a new one