# Simple question

• Oct 3rd 2011, 02:22 PM
dwsmith
Simple question
Generators of $\displaystyle \mathbb{Z}$ are $\displaystyle <1> \ \text{and} \ <-1>$ correct?
• Oct 3rd 2011, 02:38 PM
alexmahone
Re: Simple question
Quote:

Originally Posted by dwsmith
Generators of $\displaystyle \mathbb{Z}$ are $\displaystyle <1> \ \text{and} \ <-1>$ correct?

No! Generators of $\displaystyle \mathbb{Z}$ are $\displaystyle 1$ and $\displaystyle -1$.
• Oct 3rd 2011, 02:45 PM
dwsmith
Re: Simple question
Quote:

Originally Posted by alexmahone
No! Generators of $\displaystyle \mathbb{Z}$ are $\displaystyle 1$ and $\displaystyle -1$.

No? I just listed 1 and -1 too.
• Oct 3rd 2011, 02:47 PM
alexmahone
Re: Simple question
Quote:

Originally Posted by dwsmith
No? I just listed 1 and -1 too.

You listed $\displaystyle <1> \ \text{and} \ <-1>$.
• Oct 3rd 2011, 02:51 PM
dwsmith
Re: Simple question
Quote:

Originally Posted by alexmahone
You listed $\displaystyle <1> \ \text{and} \ <-1>$.

Generating set of a group - Wikipedia, the free encyclopedia
• Oct 3rd 2011, 02:57 PM
alexmahone
Re: Simple question
$\displaystyle <n>$ denotes the subgroup generated by $\displaystyle n$.

So, $\displaystyle <1>=<-1>=\mathbb{Z}$

It doesn't make sense to say that the generators of $\displaystyle \mathbb{Z}$ are $\displaystyle \mathbb{Z}$ and $\displaystyle \mathbb{Z}$, which is what you essentially did.
• Oct 3rd 2011, 03:00 PM
dwsmith
Re: Simple question
Quote:

Originally Posted by alexmahone
$\displaystyle <1>$ is the subgroup generated by $\displaystyle 1$, which is $\displaystyle Z^{+}$. Similarly, $\displaystyle <-1>$ is the subgroup generated by $\displaystyle -1$, which is $\displaystyle Z^{-}$.

It doesn't make sense to say that the generators of $\displaystyle \mathbb{Z}$ are $\displaystyle Z^{+}$ and $\displaystyle Z^{-}$, which is what you essentially did.

"The integers under addition are an example of an infinite group which is finitely generated by both <1> and <−1>"

Paragraph 1 after the contents
• Oct 3rd 2011, 03:07 PM
alexmahone
Re: Simple question
Quote:

Originally Posted by dwsmith
"The integers under addition are an example of an infinite group which is finitely generated by both <1> and <−1>"

Paragraph 1 after the contents

Not any more. (I've fixed that error now.)

In general, Wikipedia is not a good place to learn maths from.
• Oct 3rd 2011, 03:27 PM
alexmahone
Re: Simple question
• Oct 3rd 2011, 03:33 PM
dwsmith
Re: Simple question
Quote:

Originally Posted by alexmahone

http://www.csus.edu/indiv/z/zhongk/chap4hw.pdf

number 23
• Oct 3rd 2011, 03:40 PM
alexmahone
Re: Simple question
Quote:

Originally Posted by dwsmith

Number 23 says that $\displaystyle <-1>=<1>=\mathbb{Z}$, which is what I said in post #6.