# distinct generators of a group $G$

By saying distinct generator of a group $G$, is this saying that elements produced by the generator differ from elements produced by other generators?

The distinct generators of $G$ are the elements $g^r$ where $1 ≤ r < N$ and $gcd(r,N) = 1$. Thus, there are $ϕ(N)$ of them, where $ϕ$ is Euler’s phi function.

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It would be better if you would include the whole context in your question. Then we could say for sure what it meant. Try quoting the sentence from your book that you are unsure of. – MJD Aug 18 '12 at 1:57
No, it is saying that these are distinct elements of the group, with the property that each of them, individually, generates the group. – Andrew Aug 18 '12 at 2:05
@Andrew So when these groups are union-ed, it forms the group $G$? – Lucy Zeo Aug 18 '12 at 2:08
@LucyZeo, you don't have to take a union. For example $\mathbb Z/4\mathbb Z = \langle 1\rangle$, but also $\mathbb Z/4\mathbb Z = \langle -1\rangle,$ and $-1 = 3\neq 1.$ – Andrew Aug 18 '12 at 2:15
what about the second question? – Lucy Zeo Aug 18 '12 at 2:36

## 1 Answer

I'm willing to bet that in this question, $G$ is a cyclic group of order $N$. To that end, I'm answering the question I think you meant to ask. If I'm wrong, please let me know. In the case where we have a cyclic group of order $N$, there are $\varphi(N)$ different generators, each of the form of the question.

To answer your question: No. When we say an element (say $g$) generates a group, it means that every element of the group is a power of that element (of the form $g^n$ for some $n$). If we were to say that the two elements $g,h$ generate $G$, often written $G = \langle g,h \rangle$, we mean that any element of $G$ can be written as $g^n h^m$ for some $m,n$.

So any two generators generate the same set of elements. By distinct generators, we mean that the two generators are different from eachother. For example, if we are working with the cyclic group of order $2$, there are two elements. We might call them $e$ (the identity) and $g$, so that $g^2 = e$. But $g^3$ also generates this group. But $g^3$ is the same as $g$, so they are not distinct even though the 'words' describing them are not the same.

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