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!
Name for a member in a group that can cover all members by all of its exponentials?
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$\begingroup$
group-theory
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2When 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
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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.
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0@B.D, yes good point, I should have mentioned that! – 2012-11-20