"Prove that the 3-cycles in form a single conjugacy class"
Okay, I understand what a conjugacy class is (subset in which every element is conjugate to every other element in the conjugacy class), but I'm wondering exactly how I should go about proving this? Do I have to brute force my way, or is there some theorem or something I am simply forgetting about?
Okay, just to make sure I have the right idea, I could take a 3-cycle such as (3 4 5) and then multiply it with (1 2) and it would commute since both cycles share no common numbers, and thus I prove that the 3-cycles make a conjugacy class in A_5 as per the theorem you provided, correct?