= Generator of a group
= Generators of a group
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= Group generator
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A <generator of a group> is a single element $g$ for which $G=\langle g\rangle$: every <group element> is an <integer> power of $g$. A <group> admitting such an element is a <cyclic group>. In additive notation, every element must be an <integer> multiple of $g$.
Distinguish a single <generator of a group> from a <generating set of a group>. The <Klein four-group> needs two elements to generate it, although it has only four elements; no single element visits all four. A <cyclic group> need not have a unique <generator of a group>: $\mathbb Z/6\mathbb Z$ has $1$ and $5$, while the additive <integers> have $1$ and $-1$.
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