Question: Given an alternating group $A_8$, find a cyclic group with order 4 and a non-cyclic group with order 4.
So I don't know how to start this one since I'm confused how it could have a cyclic group with order 4 when it would result to three transpositions, making it an odd permutation.
Regarding the non-cyclic group, I also have no idea. Do I just map out an element to another element randomly? Or is there a technique for this?
I'd appreciate for any enlightenment for this one.