# Thread: Set of Generators and Relations

1. ## Set of Generators and Relations

Find a set of generators and relations for Z/nZ (integers modulo n).

I think it is <a, b | a^n = b^n = 1, ab=ba> but I am not positive. Would this be correct?

2. ## Re: Set of Generators and Relations

Originally Posted by letitbemww
Find a set of generators and relations for Z/nZ (integers modulo n).

I think it is <a, b | a^n = b^n = 1, ab=ba> but I am not positive. Would this be correct?
No - this group is two-generated while $\displaystyle \mathbb{Z}/n\mathbb{Z}$ is cyclic. Basically, you are looking for a one-generator group with the property that the generator to the power n is the identity. You do not actually need the commutator ab=ba there - this follows from the fact that your group is cyclic. So, plug these two stipulations into a presentation and see what you get...

3. ## Re: Set of Generators and Relations

Oh you're right I got mixed up. So it's just <a | a^n=1> ?

4. ## Re: Set of Generators and Relations

Originally Posted by letitbemww
Oh you're right I got mixed up. So it's just <a | a^n=1> ?
Yes, this is the presentation of a cyclic group of order $\displaystyle n$.

5. ## Re: Set of Generators and Relations

Originally Posted by letitbemww
Oh you're right I got mixed up. So it's just <a | a^n=1> ?
Indeed. Your original group was $\displaystyle \mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ (the cross product of two cyclic groups of order n). Can you see why?