Abstract Algebra | Cyclic Subgroups

We define the notion of a cyclic subgroup and give a few examples.

[trythisize]

In the example of U(8), since it is defined as the set of integers which are relatively prime to 8, why are 9, 11, 13, ... not included? I don't think it was ever stipulated that they need to be less than or equal to 8 to be included. He mentions something about 9, 1 being equivalent but I don't see how they're equivalent unless we're talking about equivalence mod 8, which we're not.

[vinvic]

Really interesting video !

Does the name "cyclic" come from the fact that the groups are cycles when they are subgroups of a modulo group ? Or is there a deeper reason ?