Howdy all, having issues with some group theory problems. Hints or suggestions are very welcomed.
(1) Prove that for n>=3 the subgroup generated by the three cycles is An.
(2) Prove that the smallest subgroup of Sn containing (1, 2) and (1, 2, . . . , n) is Sn
This is what I have so far:
Sn is generated by 2-cycles. Any element in An is a product of an even number of transpositions, so we only need to show the product of any two transpositions can be written in terms of 3-cycles.
(12)(13) = (123), while
(12)(34) = (123)(143).
Is this sufficient enough, I feel as if this is not proofy. HELP!!