Cyclic Order - Finite Cycles

Finite Cycles

A cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Each element x in X has a "next element" and a "previous element", and taking either successors or predecessors cycles exactly once through the elements as x(1), x(2), ..., x(n). In other words, a cyclic order on X is the same as a permutation that makes all of X into a single cycle.

A cycle with n elements can be even more succinctly defined as a Z_n-torsor: a set with a free transitive action by a finite cyclic group. Another way to put it is to say that we make X into the standard directed cycle graph on n vertices, by some matching of elements to vertices.

It can be instinctive to use cyclic orders for symmetric functions, for example as in

xy + yz + zx

where writing the final monomial as xz would distract from the pattern.

A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements g and h of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and y−1 with y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rules that allow one to remove or add adjacent y and y−1.

A cyclic order on a set X can be determined by a linear order on X, but not in a unique way. Choosing a linear order is equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order. Since there are n! possible linear orders, there are (n − 1)! possible cyclic orders.