Recipe

Find every generator of a finite cyclic group

Use when: you have a finite cyclic group of order $n$ and need every element that generates it, not just one

Goal: the complete, exact list of generators, or the conclusion that the group has none

Taught in Generators of a cyclic group

Quick recipe

Compute the order of every non-identity element
Compare each order to the group's own order n
Verify a claimed generator by writing out its orbit

1
For each element $g$ in the group, find $|g|$ by combining $g$ with itself until reaching $e$ , the same method the last section used.

Example

In $\mathbb{Z}_6$ : $|1|=6$ , $|2|=3$ , $|3|=2$ , $|4|=3$ , $|5|=6$ .

Why: a generator is exactly an element whose own orbit already reaches every position in the group.
2
An element $g$ is a generator exactly when $|g|=n$ ; anything smaller means $g$ 's orbit only reaches part of the group.

Example

$\mathbb{Z}_6$ has $n=6$ ; among the orders from Step 1, only $1$ and $5$ equal $6$ , so those are exactly the generators.
- some element's order equals $n$ — that element is a generator; keep checking the rest for others.
- every element's order is strictly less than $n$ — the group has no generator at all.
⚡ Shortcut

Inside $\mathbb{Z}_n$ specifically, a generator can be spotted without computing a single order: $k$ generates $\mathbb{Z}_n$ exactly when $k$ shares no factor with $n$ besides $1$ . In $\mathbb{Z}_6$ , checking shared factors directly, $1$ and $5$ share none with $6$ , while $2$ , $3$ , $4$ each share a factor with $6$ , the same two generators the order computation found.

Why: this is a direct restatement of the definition of generator, applied element by element instead of guessed at.
3
Pick one confirmed generator and list its first several powers, to confirm they actually cover the whole group.

Example

Check $5$ in $\mathbb{Z}_6$ : $5, 5+5=10\equiv4, 4+5=9\equiv3, 3+5=8\equiv2, 2+5=7\equiv1, 1+5=6\equiv0$ , visiting all six elements.

Why: matching orders is reliable, but writing out the actual orbit once catches any bookkeeping slip made in Step 1.

Common mistakes

Assuming every non-identity element is a generator just because it isn't the "obviously wrong" choice.
Stopping the search after finding one generator instead of checking every element for the full list.
Using the $\mathbb{Z}_n$ shared-factor shortcut inside a group that isn't $\mathbb{Z}_n$ , where it doesn't apply.
Reporting the group's own order when asked for a specific element's order, or vice versa.