Cycle Index - Types of Actions

Types of Actions

As the above example shows, the cycle index depends on the group action and not on the abstract group. Since there are many permutation representations of an abstract group, it is useful to have some terminology to distinguish them.

When an abstract group is defined in terms of permutations, it is a permutation group and the group action is the identity homomorphism. This is referred to as the natural action.

The symmetric group S₃ in its natural action has the elements

and so, its cycle index is:

$Z(S_3) = \frac{1}{6} \left( a_1^3 + 3 a_1 a_2 + 2 a_3 \right).$

A permutation group G on the set X is transitive if for every pair of elements x and y in X there is at least one g in G such that y = xg. A transitive permutation group is regular (or sometimes referred to as sharply transitive) if the only permutation in the group that has fixed points is the identity permutation.

A finite transitive permutation group G on the set X is regular if and only if |G| = |X|. Cayley's theorem states that every abstract group has a regular permutation representation given by the group acting on itself (as a set) by (right) multiplication. This is called the regular representation of the group.

The cyclic group C₆ in its regular representation contains the six permutations (one-line form of the permutation is given first):

= (1)(2)(3)(4)(5)(6)

= (1 2 3 4 5 6)

= (1 3 5)(2 4 6)

= (1 4)(2 5)(3 6)

= (1 5 3)(2 6 4)

= (1 6 5 4 3 2).

Thus its cycle index is:

$Z(C_6) = \frac{1}{6} \left( a_1^6 + a_2^3 + 2 a_3^2 + 2 a_6 \right).$

Often, when an author does not wish to use the group action terminology, the permutation group involved is given a name which implies what the action is. The following three examples illustrate this point.

Famous quotes containing the words types of, types and/or actions:

“The wider the range of possibilities we offer children, the more intense will be their motivations and the richer their experiences. We must widen the range of topics and goals, the types of situations we offer and their degree of structure, the kinds and combinations of resources and materials, and the possible interactions with things, peers, and adults.”
—Loris Malaguzzi (1920–1994)

“The rank and file have let their servants become their masters and dictators.... Provision should be made in all union constitutions for the recall of leaders. Big salaries should not be paid. Career hunters should be driven out, as well as leaders who use labor for political ends. These types are menaces to the advancement of labor.”
—Mother Jones (1830–1930)

“A young man is not a proper hearer of lectures on political science; for he is inexperienced in the actions that occur in life, but its discussions start from these and are about these; and, further, since he tends to follow his passions, his study will be vain and unprofitable, because the end that is aimed at is not knowledge but action. And it makes no difference whether he is young in years or youthful in character.”
—Aristotle (384–323 B.C.)