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.

Read more about this topic:  Cycle Index

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

    “Our children evaluate themselves based on the opinions we have of them. When we use harsh words, biting comments, and a sarcastic tone of voice, we plant the seeds of self-doubt in their developing minds.... Children who receive a steady diet of these types of messages end up feeling powerless, inadequate, and unimportant. They start to believe that they are bad, and that they can never do enough.”
    —Stephanie Martson (20th century)

    “... there are two types of happiness and I have chosen that of the murderers. For I am happy. There was a time when I thought I had reached the limit of distress. Beyond that limit, there is a sterile and magnificent happiness.”
    —Albert Camus (1913–1960)

    “The mob is man voluntarily descending to the nature of the beast. Its fit hour of activity is night. Its actions are insane like its whole constitution. It persecutes a principle; it would whip a right; it would tar and feather justice, by inflicting fire and outrage upon the houses and persons of those who have these. It resembles the prank of boys, who run with fire-engines to put out the ruddy aurora streaming to the stars.”
    —Ralph Waldo Emerson (1803–1882)