Elements of Subgroups in Additive Group Q/Z: G(2) and G(P)

[Thorn]

The question:

If G is the additive group Q/Z, what are the elements of the subgroup G(2)? Of G(P) for any positive prime P?

Where G(n)={a e G| |a| = n^(k) for some k is greater than or equal to 0}...That is the set of all a in G, s.t. the order of a is some power of n. (But since it is the additive group, I suppose it would just a be a multiple of n)

How do I even begin with this? Aren't the elements of Q/Z sets? The collections of right cosets? and don't they have infinite order?...

 

[morphism]

(But since it is the additive group, I suppose it would just a be a multiple of n)

I don't see why you would suppose that. The additive notation has no bearing on this matter.

Aren't the elements of Q/Z sets? The collections of right cosets?

Sure.

and don't they have infinite order?...

No. Take an element r+Z in Q/Z. What does it mean for n(r+Z) to be the zero element of Q/Z? If you can answer this correctly, then you can easily deduce that all the elements of Q/Z have finite order.

How do I even begin with this?

My suggestion is to actually think about what Q/Z is, as a group. Make sure you understand how the group operation works, and what order means in this setting.