"Question: If I have where , is it correct to say that . If so, then contains elements. Must all the elements be distinct? Must an identity be in the group"
If you mean that G is a cyclic group (in other words, the entire group is ) such that is as high as you can get without repeating, then actually because has 5 elements (count them).
Yes, by definition, a group must have an identity, which in this case is , and all elements must be distinct. For example, although and look different, they're actually the same element because in a cyclic group, the elements start ``repeating.''