(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let G be a cyclic group of order n, and let r be an integer dividing n. Prove that G contains exactly one subgroup of order r.

2. Relevant equations

cyclic group, subgroup

3. The attempt at a solution

Say the group G is {x^0, x^1, ..., x^(n-1)}

If there is a subgroup H of order r, it must be cyclic, because: why? I can't figure it out, but I have a feeling that it must be cyclic.

H is generated by some element, call it b=x^m. Since x^r = 0, we have (x^m)(x^m)... (r times) = 0. Thus mr=n and H must be the cyclic group generated by x^(n/r).

I have a feeling that I have the right idea but I don't know how to show that a group is cyclic. Could someone help?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Cyclic group/ subgroup proof

**Physics Forums | Science Articles, Homework Help, Discussion**