Exercise · ZW-CYCLIC-GROUP-009 · ★★★ Advanced

Let \(G=\langle g\rangle\) be an infinite cyclic group. How many generators does \(G\) have?

Choose one option.

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Explanation

The generators are \(g\) and \(g^{-1}\). Any other power \(g^k\) with \(|k|\ge 2\) misses \(g\), since \(g=(g^k)^m\) would force \(km=1\).

A cyclic group is a group \(G\) with an element \(g\), called a generator, such that every element of \(G\) is a power of \(g\).

\[G=\langle g\rangle=\{\,g^k : k\in\mathbb{Z}\,\}\]