Simple question

**dwsmith** · October 3rd 2011, 02:22 PM

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

**alexmahone** · October 3rd 2011, 02:38 PM

Originally Posted by dwsmith

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

No! Generators of $\mathbb{Z}$ are $1$ and $-1$ .

**dwsmith** · October 3rd 2011, 02:45 PM

Originally Posted by alexmahone

No! Generators of $\mathbb{Z}$ are $1$ and $-1$ .

No? I just listed 1 and -1 too.

**alexmahone** · October 3rd 2011, 02:47 PM

Originally Posted by dwsmith

No? I just listed 1 and -1 too.

You listed $<1> \ \text{and} \ <-1>$ .

**dwsmith** · October 3rd 2011, 02:51 PM

Originally Posted by alexmahone

You listed $<1> \ \text{and} \ <-1>$ .

Generating set of a group - Wikipedia, the free encyclopedia

**alexmahone** · October 3rd 2011, 02:57 PM

Originally Posted by dwsmith

Generating set of a group - Wikipedia, the free encyclopedia

$<n>$ denotes the subgroup generated by $n$ .

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

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

**dwsmith** · October 3rd 2011, 03:00 PM

Originally Posted by alexmahone

$<1>$ is the subgroup generated by $1$ , which is $Z^{+}$ . Similarly, $<-1>$ is the subgroup generated by $-1$ , which is $Z^{-}$ .

It doesn't make sense to say that the generators of $\mathbb{Z}$ are $Z^{+}$ and $Z^{-}$ , 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

**alexmahone** · October 3rd 2011, 03:07 PM

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.

**alexmahone** · October 3rd 2011, 03:27 PM

See Example 1.4 of http://www.math.uconn.edu/~kconrad/b...ory/genset.pdf.

**dwsmith** · October 3rd 2011, 03:33 PM

Originally Posted by alexmahone

See Example 1.4 of http://www.math.uconn.edu/~kconrad/b...ory/genset.pdf.

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

number 23

**alexmahone** · October 3rd 2011, 03:40 PM

Originally Posted by dwsmith

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

number 23

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

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