I fear I have become stumped with permutations once we hit cycles. I understand cycle notation, and multiplication. I desperately seek help on a few homework problems that I just cannot find footing on!

1. Show that A8 contains an element of order 15.

I thought that since n=8 the largest element could be 8?

2. Prove that (1234) is not the product of 3-cycles.

3. In S4, find a cyclic subgroup of order 4 and a noncyclic subgroup of order 4.

I really do not know where to start on these, the book is quite unhelpful.

Thank you!