Im taking abstract algebra right now and it took me a long time to notice the point of cosets: we use the fact that aH is merely a permutation of the elements of H (assuming H is a subgroup and a is an element of H) to conclude aH = H (since sets are not ordered, permuting the elements doesnt change anything). So if we want to treat permutations of H as H itself, we work with cosets.
However, if H was instead an ordered tuple containing every element of H, aH would not be the same as H because order matters now. So this time, aH would be a sort of "ordered coset". Is that a thing? Or is that merely a different way to write aH as a permutation of H?
I came up with this question when solving a simple cipher where you merely shift the alphabet by a certain amount. I noticed that a shift of, say 5, heavily reminds me of $5 + \mathbb{Z}_{26}$ (or 5 + H). Then I noticed that this idea fails because $5 + \mathbb{Z}_{26} = \mathbb{Z}_{26}$ (its merely a permutation). If this idea is doomed to fail, are there other ways to understand this simple cypher in terms of algebra? Permutations maybe?