## Cyclic Groups

Sometimes it is the case that a group is generated by repeatedly composing a single elementwith itself. In this case, the group is said to be a cyclic group generated by the element

Every rotation group of orderconsisting of the rotations of the regular polygon withsides. The elementmay be taken to be the element that rotates the polygon byWe writemeaning that G is generated by the element

We may writeit terms of its distinct elements, all form by composition of a with itself.

Every elepment ofis found by repeated compositions ofwith itself.

An elemtentthat generates a cyclic group of ordermust have order sothe identity element, andfor any

Every subgroup of a cyclic group is also cyclic, and the elements of each cyclic subgroup of a cyclic group with generatorof orderis formed by repeated compositions ofwith itself, for someIf the subgroup generated byhas order s, thenso that

The Cayley table for the rotation symmetries of a regular hexagon is given below.

Notice that each row is displaced one to the left of the row above it and wrapped, so that the leftmost element reappears on the right.