Applied finite group actions. 2nd, rev. and exp. ed.

*(English)*Zbl 0951.05001
Algorithms and Combinatorics. 19. Berlin: Springer. xxvi, 454 p. (1999).

This second edition of an introduction to the use of finite group actions in combinatorics extends the classical approach of only counting objects in the first edition to the more demanding task of constructing objects. The theory of counting unlabelled objects is carefully developed. The objects are introduced via the theory of species, counting is based on the Cauchy-Frobenius lemma, Redfield’s and Pólya’s theory, Burnside’s table of marks and Plesken’s extension of Burnside rings. Applications, especially to mathematical chemistry highlight the approach. Also, the representation theory of symmetric groups in incorporated, leading to the practical use of symmetry adapted bases. The methods for constructing objects up to isomorphism are emerging from the use of computers in combinatorics. The book contains some fairly general methods: orderly generation, use of homomorphisms, double cosets and some random generation. Some examples are chosen from recent research on chemical structure elucidation and \(t\)-designs. Applications to coding theory can be found in another book by the author and his school [Codierungstheorie (Springer, Berlin) (1998; Zbl 0922.94009)]. The present book may well serve as an advanced introduction into the subject. In many aspects it leads to the present state of the art. There is a collection of material on character tables of symmetric groups. Some historical remarks and suggestions for further reading conclude the book.

Reviewer: Reinhard Laue (Bayreuth)

##### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05E20 | Group actions on designs, etc. (MSC2000) |

20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |

20B30 | Symmetric groups |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

05E30 | Association schemes, strongly regular graphs |

05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |