# Thread: Generators for Z/nZ

1. ## Generators for Z/nZ

Dummit and Foote Section 1.2 Dihedral Groups Exercise 15 reads as follows:

Find a set of generators and relations for $\displaystyle \mathbb{Z}$/n$\displaystyle \mathbb{Z}$
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If you particularize the problem to say $\displaystyle \mathbb{Z}$/4$\displaystyle \mathbb{Z}$ then 1 + 4$\displaystyle \mathbb{Z}$ is a generator and so is 3 + $\displaystyle \mathbb{Z}$.

But relations??? Are there any?

I guess then 1 + n$\displaystyle \mathbb{Z}$ is a generator for $\displaystyle \mathbb{Z}$/n$\displaystyle \mathbb{Z}$. Is that correct? Other generators? Relations???

Can someone please clarify and help?

Peter

2. ## Re: Generators for Z/nZ

Z/nZ is cyclic, so it has at least one generator, call it x. we only need to subject this generator to one relation to recover all the algebraic behavior of Z/nZ:

x^n = e.

(this is a really simple example, so don't over-think it).

3. ## Re: Generators for Z/nZ

Thanks

You write "this is a really simple example, so don't over-think it"

Good advice - I was looking for too much in it

Peter

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