Comments and References
This chapter begins with remarks on the history of finite group actions. The reader will then find comments that point to certain important articles and books, together with hints for further reading and additional references.
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This chapter begins with remarks on the history of finite group actions. The reader will then find comments that point to certain important articles and books, together with hints for further reading and additional references.
12.1 Historical Remarks, Books and Review Articles Chapter 0 of the present book is devoted to labeled structures. They are important since unlabeled structures, which are the main subject here, are equivalence classes of labeled ones. In order to give a flavor of a modern treatment of labeled structures, too, the first chapter contains the basics of the theory of species. It should serve as an appetizer for the standard book on this topic, which is the book [7], by Bergeron, Labelle and Leroux. The original paper was by Joyal ([73]). Another important paper on species theory and it applications is the Habilitationsschrift of Strehl ([149]). The chapter 1 is mainly devoted to the introduction of unlabeled structures as orbits of finite groups on finite sets. The enumeration of such structures, which is described in chapter 3, can be done by an application of the Cauchy-Frobenius Lemma, the history of which is described in articles by Neumann ([ 11 0]) and Wright ([168]). This lemma is mostly ascribed to Burnside, who gives it in the first edition of his book on finite groups ( l21] ), but the lemma is contained in section 118, while the ascription is at the beginning of section 119. In the second edition of this book ([20], 1911, reprinted by Dover Publications in 1955) which is mostly quoted, these sections are completely rewritten, and Burnside omits the ascription. This might be the reason for usually attributing this lemma to Burnside. Burnside's reference is to [55], a paper of Frobenius. Frobenius gives credit to Cauchy who proved this lemma for the transitive case in [34]. In fact Burnside proved a much stronger result, which is described in chapter 4 of the present book. Besides the basic concepts of the theory of finite group actions, the second chapter contains in particular the notion of symmetry classes of mappings and the corresponding enumerative results. The pioneering publication on this topic was the famous paper [117] by P6lya, a masterpiece. It had a predecessor ([125]), a paper by Redfield, which was overlooked for many years. In fact Redfield's paper contains stronger results than P6lya's, but it is very difficult to read since it expresses the results in terms of operations on polynomials that can be understood more or less only in terms of linear representation theory. There was at least one further paper written A. Kerber, Applied Finite Group Actions © Springer-Verlag Berlin Heidelberg 1999
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12. Comments and References
by Redfield, it had been rejected once but it was published recently ([124]). Another paper, entitled "Enumeration distinguishable arrangements for general frame groups", was found together with an untitled manuscript. They are not published yet. A translation of P6lya's paper into English, together with an article on the fifty years' history of
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