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

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