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Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
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Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.
COUNTING WITH SYMMETRIES
"... Since I have not found any literature I really like on this subject, these notes are supposed to help you. If you prefer a book I would recommend [1] (which is in the reserve in Millikan) as text. It should be noted that we will consider only finite groups and sets. The essential result (everything ..."
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Since I have not found any literature I really like on this subject, these notes are supposed to help you. If you prefer a book I would recommend [1] (which is in the reserve in Millikan) as text. It should be noted that we will consider only finite groups and sets. The essential result (everything we will discuss is either a direct corollary of this result or a slight generalization) is the following result by Frobenius: Theorem (Burnside’s Lemma). Suppose a finite group G acts on a finite set X. Then the number of orbits g∈G Xg, where Xg = {x ∈ X: gx = x}. in X is
Communicated by the Managing Editors
, 1979
"... In this paper we apply Polya’s theory of counting to compute the number of isomorphism types of reduced Witt rings of fields with a fixed finite number of orderings. The problem is first transformed into a graph theoretical enumeration problem involving unlabeled rooted trees with certain numbers a ..."
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In this paper we apply Polya’s theory of counting to compute the number of isomorphism types of reduced Witt rings of fields with a fixed finite number of orderings. The problem is first transformed into a graph theoretical enumeration problem involving unlabeled rooted trees with certain numbers assigned to the vertices. The purpose of this paper is to count the number of isomorphism types of reduced Witt rings of nondegenerate symmetric bilinear forms over fields with a fixed number of orderings. All of the background material necessary for understanding the situation in which this counting problem arises will be described in Section 1. Using results from [2], this problem is restated in Section 2 in terms of an equivalent counting problem in graph theory. More specifically, it involves counting all unlabeled rooted trees which have numbers assigned to their vertices satisfying certain conditions. A great deal of work has been done previously by several authors in solving graph enumeration problems [3 1. The techniques we use in Section 2 to solve our counting problem will be entirely combinatorial and can be thought of as an application of these earlier enumeration techniques to a problem arising in quadratic form theory. 1. BACKGROUND MATERIALIN QUADRATIC FORMS Let F be ‘an arbitrary field of characteristic different from 2. In studying the algebraic theory of quadratic forms, one finds it very useful to form a ring W(F) called the Witt ring of the field. Its elements consist of