Cycles
Cycles can overlap, or they can have no element in common but the identity. The cycle graph displays each interesting cycle as a polygon.
If a generates a cycle of order 6 (or, more shortly, has order 6), then a6 = e. Then the set of powers of a², {a², a4, e} is a cycle, but this is really no new information. Similarly, a5 generates the same cycle as a itself.
So, we only need to consider the primitive cycles, those that are not subsets of another cycle. Each of these is generated by some primitive element, a. Take one point for each element of the original group. For each primitive element, connect e to a, a to a², ... an-1 to an, ... until you come back to e. The result is the cycle graph.
(Technically, the above description implies that if a² = e, so a has order 2 (is an involution), it's connected to e by two edges. It's conventional to only use one.)
Read more about this topic: Cycle Graph (algebra)
Famous quotes containing the word cycles:
“The stars which shone over Babylon and the stable in Bethlehem still shine as brightly over the Empire State Building and your front yard today. They perform their cycles with the same mathematical precision, and they will continue to affect each thing on earth, including man, as long as the earth exists.”
—Linda Goodman (b. 1929)