Applications and Related Concepts of Orbit-Stabilizer Theorem

Xien Dou

doi:10.54097/9vy1pn16

Authors

Xien Dou

DOI:

https://doi.org/10.54097/9vy1pn16

Keywords:

Orbit-Stabilizer Theorem, Group Action, Class Equation, Burnside’s Lemma.

Abstract

As the essential part of abstract algebra, group theory takes a critical place on the field of pure mathematics. Approaching the study of it from the aspect of group action is a great starting point. This paper will firstly expound several related concepts and theorems, such as stabilizer, orbit, fixator, conjugacy class, center of group and centralizer, and the Lagrange’s Theorem. Based on that, the paper investigates three essential theorems themselves and their proofs, which are Orbit-Stabilizer Theorem, Class Equation, and Burnside’s Lemma. This paper specifically provides wonderful insights about the applications of the Orbit-Stabilizer Theorem. The theorem is directly utilized in the derivations of Burnside’s Lemma and the Class Equation, also in the deduction of the general formula of the order of symmetric group . There are always some connections between the concepts in these theorems and examples with the orbit and stabilizer, thus Orbit-Stabilizer Theorem can perform brilliantly. This paper provides wonderful insights that effectively connect the Orbit-Stabilizer Theorem with other parts of Group Theory, also brings some new ideas to solving problems and prove theorems.

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