Composing group action to permutation representation
- 1 Shandong University
* Author to whom correspondence should be addressed.
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 Burnside’s 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 Burnside’s lemma in another way by permutation representation.
Keywords
Group Action, Orbit-Stabilizer Theorem, Burnside’s Lemma, Permutation Representation
[1]. Dummit D S and Foote R M 2004 Group Actions. in Abstract Algebra (3rd ed.), New York, Wiley, 41.
[2]. Eie M and Chang S T 2010 A Course on Abstract Algebra. Iowa, World Scientific, 253.
[3]. Hasselbarth W 1986 An Invitation to Permutation Representations of Groups. Croatica Chemica Acta, 59(3) 565-582.
[4]. Burnside W 2017 Theory of groups of finite order (2nd ed). New York: Dover Publications, Inc.Mineola.
[5]. Smith J D 2006 Permutation representations of left quasigroups. Algebra universalis, 55 387-406.
[6]. Steinberg B 2011 Permutation Representation. in Representation Theory of Finite Groups: An Introductory Approach, Netherlands, Springer Nature, 85-88.
[7]. Cherniavsky Y and Sklarz M 2008 Conjugacy in permutation representations of the symmetric group. Communications in Algebra, 1726-1738.
[8]. Cherniavsky Y and Bagno E 2006 Permutation representations on invertible matrices. Linear algebra and its applications, 494-518.
[9]. Serre J P 1977 Generalies on linear representation. Linear representation, New York, Springer.
[10]. Artin M 2011 The Operation on Cosets. in Algebra (2nd ed.), Boston, Pearson Prentice Hall, 179.
Cite this article
Liu,Z. (2023). Composing group action to permutation representation. Theoretical and Natural Science,9,30-37.
Data availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
Disclaimer/Publisher's Note
The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of EWA Publishing and/or the editor(s). EWA Publishing and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
About volume
Volume title: Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and
conditions of the Creative Commons Attribution (CC BY) license. Authors who
publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons
Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this
series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published
version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial
publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and
during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See
Open access policy for details).