Permutation As Product of Cycles
Let be a permutation of, and let
be the orbits of with more than 1 element. Consider an element, let denote the cardinality of, =. Also, choose an, and define
We can now express as a product of disjoint cycles, namely
Since disjoint cycles commute with each other, the meaning of this expression is independent of the convention used for the order in products of permutations, namely whether the factors in such a product operate rightmost-first (as is usual more generally for function composition), or leftmost-first as some authors prefer. The meaning of individual cycles is also independent of this convention, namely always as described above.
Read more about this topic: Cycle Notation
Famous quotes containing the words product and/or cycles:
“For man is not the creature and product of Mechanism; but, in a far truer sense, its creator and producer.”
—Thomas Carlyle (1795–1881)
“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)