Composing group action to permutation representation

Abstract

A mapping that satisfies two specific axioms provides a common notion of group action. A homomorphism translating from a group to a symmetric group of a certain set can also be used to describe group action. Therefore, any example of the group actions can be stated based on the second equivalent definition, such as the regular action, natural matrix action, coset action, and Z^2 acting on R^2, etc. It is necessary to examine the concepts of the orbit and stabilizer of a group in order to reveal the orbit-stabilizer theorem. After the preparatory work, the orbit-stabilizer theorem can be proved by defining a mapping from the orbit to the stabilizer and then checking that the mapping is well-defined and bijective. To derive Burnsides lemma, it needs to introduce the set of fixed points which is related to the concept of the stabilizer. Through the orbit-stabilizer theorem along with the fact that a set is a disjoint union of orbits, Burnside's lemma can be confirmed. Moreover, it is natural to compose a group action with a linear representation, and then a representation would be obtained, which is permutation representation. Further, one must calculate the character of the permutation representation, the dimension of the fixed subspace, and the dimension of CX^G. Then it can show Burnsides lemma in another way by permutation representation.