Exploring Subgroups of the Additive Group Q/Z in Abstract Algebra

[Thorn]

Homework Statement

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)

Homework Equations

The Attempt at a Solution

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?...

This might be a more appropriate place for this question...

 

[mighty2000]

First, let's clarify some terminology. In the context of group theory, the term "sets" usually refers to the elements of a group, not the group itself. So in this case, the elements of Q/Z would be the rational numbers modulo 1 (i.e. fractions with denominator 1). The group itself is the collection of these elements under the operation of addition.

Now, to answer your question, the elements of G(2) would be the elements in G whose order (i.e. the number of times you have to add it to itself to get the identity element) is a power of 2. In this case, since G is the additive group, the order of an element is simply the multiple of 2 that it represents. So G(2) would consist of elements like 1/2, 3/2, 5/2, etc.

Similarly, for G(P), the elements would be those whose order is a power of P. So G(3) would consist of elements like 1/3, 2/3, 4/3, etc.

I hope this helps clarify things for you. If you have any further questions, please don't hesitate to ask. Good luck with your studies!