Actions of Groups, Topological Groups and Semigroups

Mahima Ranjan Adhikari and Avishek Adhikari
Additional contact information
Mahima Ranjan Adhikari: Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)
Avishek Adhikari: University of Calcutta, Department of Pure Mathematics

Chapter Chapter 3 in Basic Modern Algebra with Applications, 2014, pp 137-158 from Springer

Abstract: Abstract Chapter 3 discusses actions of semigroups, groups, topological groups, and Lie groups. Each element of a group determines a permutation on a set under a group action. For a topological group action on a topological space, this permutation is a homeomorphism and for a Lie group action on a differentiable manifold it is a diffeomorphism. Group actions are used in the proofs of the Counting Principle, Cayley’s Theorem, Cauchy’s Theorem and Sylow Theorems for finite groups. The counting principle is used to determine the structure of a finite group. These groups arise in the Sylow Theorems and in the description of finite abelian groups. The orbit spaces obtained by topological group actions, discussed in this chapter, are very important in topology and geometry. For example, n-dimensional real and complex projective spaces are obtained as orbit spaces. Finally, semigroup actions are applied to theoretical computer science yielding state machines which unify computer science with mainstream mathematics.

Keywords: State Machine; Finite Group; Finite State Machine; Orbit Space; Prime Integer (search for similar items in EconPapers)
Date: 2014
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-81-322-1599-8_3

Ordering information: This item can be ordered from
http://www.springer.com/9788132215998

DOI: 10.1007/978-81-322-1599-8_3

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().