1
$\begingroup$

In a group $G$, if there is a member $x$ of $G$ s.t. $G=\{x^n, n \in \mathbb{N} \text{ or } \mathbb{Z}\}$, is there a name for such $x$? Thanks!

  • 2
    When it is isomorphic to one of $\mathbb{Z}$ or some $\mathbb{Z}_n$. But that's just a mild restatement.2012-11-19

1 Answers 1

1

As you have seen above, such an element is called a generator.

Note, however, that if $n \in \mathbb{N} = \{0, 1, 2, 3, \ldots\}$ then $G$ would have to be a finite group. (Why?)

For this reason, it is preferable to consider groups $G$ such that $G = \{x^n | n \in \mathbb{Z}\}$ for some $x \in G$.

In such a case, $x$ is called the generator and the group is called cyclic; however, using $\mathbb{Z}$ allows you to talk about both finite and infinite cyclic groups with the same notation.

  • 0
    @B.D, yes good point, I should have mentioned that!2012-11-20