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Formal Definition

An element $a$ of a group $G$ generates $G$ and is a generator for $G$ if the cyclic group generated by $a$, denoted $\langle a \rangle$, equals $G$. A group $G$ is cyclic if there is some element a in $G$ that generates $G$.

Informal Definition

An element of group $G$, call it a, is a generator for $G$ if every element in $G$ can be made up of a combination of a.

Example(s)

Both $1$ and $3$ are generators of $\mathbb{Z}_4$, that is $\langle 1\rangle=\langle 3\rangle=\mathbb{Z}_4$.

Non-example(s)

2 cannot be a generator of $\mathbb{Z}_4$ because it will produce the set ${0,2}$, that is $\langle2\rangle\not=\mathbb{Z}_4$