Cyclic subgroup of a particular form proof with the group infinitely large.

[jessicaw]

Homework Statement

G is additive group. If the order of G is infinity, then G is cyclic iff each subgroup H of G is of the form nG for some interger n.

Homework Equations

cyclic property.

The Attempt at a Solution

i kind of know that nG is the answer but why G has to be infinite order?
And i think it is most difficult to write out the proof.


I love this forum and all the kind and great homework helpers.

 

[mighty2000]

Hello there! Thank you for your post. Let me try to explain why G needs to have infinite order for it to be cyclic.

First, let's define what it means for a group to be cyclic. A group G is cyclic if there exists an element g in G such that every element in G can be written as a power of g. In other words, every element in G can be generated by repeatedly applying the operation of the group to g.

Now, let's consider the case where G has finite order. This means that there is a finite number of elements in G. Since every element in G can be written as a power of g, this means that there is a finite number of powers of g that can generate all the elements in G. But this contradicts the fact that G has finite order. Therefore, G cannot be cyclic if it has finite order.

On the other hand, if G has infinite order, this means that there is no limit to the number of powers of g that can generate all the elements in G. In other words, every element in G can be written as a power of g, and there is no limit to the number of powers needed to generate all the elements. This aligns with the definition of a cyclic group.

Now, let's move on to the second part of the statement, which says that each subgroup H of G is of the form nG for some integer n. This is known as the cyclic property of subgroups. It means that every subgroup of G can be generated by a single element, which is a power of g. This is only possible if G has infinite order, as explained above.

To prove this statement, you can use the fact that every subgroup of a cyclic group is also cyclic. Therefore, if G is cyclic, every subgroup of G is also cyclic and can be generated by a single element.

I hope this helps clarify why G needs to have infinite order for it to be cyclic and why the cyclic property of subgroups is only possible in infinite order groups. Keep up the good work in your studies!