Generators of Cyclic Groups

Cyclic groups usually have more than one generator – at least if the group has more than one member.

The setis cyclic group under addition mod 5, generated by the element 1 but also by the elements 2, 3 and 4.

The setis a cyclic group under addition mod 6, generated by the element 1 but also by the elements 5.

Every cyclic group of orderis isomorphic to the addition group

The following theorem gives the condition under which an element of a cyclic group may generate the group.

Letbe a cyclic group of orderThenif and only if the greatest common denominator ofandis 1 ().

Proof: Ifwe can writefor integersandThenThenso all powers ofbelong tohenceandgenerates

Suppose then thatWriteandso that the order ofshowing thatdoes not generate