(no subject)

A group action from a group G to a set X is a homomorphism between G and the symmetric group over X. I wonder if there is some systematic way to enumerate all group actions from G to X.

I think I have an answer for this: G is partitioned into cosets, and each coset is identified with an element of X. Since a partition into cosets is determined by a choice of subgroup H, we can enumerate all possible group actions by enumerating subgroups of G, and for each subgroup choice H we can find the cosets, and then associate them with elements of X... implying also that |X|=|G|/|H|. Does this make sense? And, is there a requirement that H be normal?