# Generators for Z/nZ

• Oct 15th 2011, 03:48 AM
Bernhard
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
• Oct 15th 2011, 01:57 PM
Deveno
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).
• Oct 15th 2011, 04:18 PM
Bernhard
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