Sometimes it is the case that a group is generated by repeatedly composing a single element
with itself. In this case, the group is said to be a cyclic group generated by the element
Every rotation group of order
consisting of the rotations of the regular polygon withsides. The elementmay be taken to be the element that rotates the polygon by
We write
meaning that G is generated by the element
We may write
it 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
so
the identity element, and
for 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 of
with itself, for some
If the subgroup generated byhas order s, then
so that
The Cayley table for the rotation symmetries of a regular hexagon is given below.
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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.