How do I see a group's generator? For example in $\mathbb{Z}_{10}$ ? I'm reading my school book but I don't find any methods.
How do I see a group's generator?
1
$\begingroup$
linear-algebra
group-theory
cyclic-groups
-
1With an example this small, you could just compute $\langle x \rangle$ for each $x \in \Bbb Z_{10}$ and see what happens. – 2017-01-29
1 Answers
1
I suspect you are looking for generating sets - the group $\mathbb{Z}_2\oplus\mathbb{Z}_2$ for example has no single generator, but has generating sets $\{(1,0),(0,1)\}$ for example.
It is not generally immediately obvious what a generating set for a group is. In the case of cyclic groups, $\mathbb{Z}_n$, it is a nice exercise to show that $m\in\mathbb{Z}_n$ is a generator if and only if $\mathrm{gcd}(m,n)=1$. For more general groups you're better off searching for or asking about the specific group.