How about ?

No. Any can be written as a product of disjoint cycles. And is the lcm of the lengths of these cycles. Thus, the only way for to have order 6 is if is a product of a -cycle and a -cycle. However, a product of a -cycle and a -cycle is anDoes A6 have an element of order 6?oddpermutation. Thus, it is not in which hasevenpermutations.