1. ## Simple question

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

2. ## Re: Simple question

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 $\displaystyle -1$.

3. ## Re: Simple question

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

4. ## Re: Simple question

Originally Posted by dwsmith
No? I just listed 1 and -1 too.
You listed $\displaystyle <1> \ \text{and} \ <-1>$.

5. ## Re: Simple question

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

Generating set of a group - Wikipedia, the free encyclopedia

6. ## 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.

7. ## Re: Simple question

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>"

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8. ## Re: Simple question

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.

11. ## Re: Simple question

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