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As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets. In the special case that the underlying set is linearly ordered, there is a natural subgroup to study, namely the set of permutations that preserves that order. In some senses. these are universal for automorphisms of models of theories. The purpose of this book is to make a thorough, comprehensive examination of these groups of permutations. After providing the initial background Professor Glass develops the general structure theory, emphasizing throughout the geometric and intuitive aspects of the subject. He includes many applications to infinite simple groups, ordered permutation groups and lattice-ordered groups. The streamlined approach will enable the beginning graduate student to reach the frontiers of the subject smoothly and quickly. Indeed much of the material included has never been available in book form before, so this account should also be useful as a reference work for professionals.

Permutation groups. --- Ordered groups. --- Group theory --- Substitution groups --- Permutation groups --- Ordered groups

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Groupes, Théorie des. --- Group theory. --- Groupes de permutations. --- Permutation groups.

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These notes present an investigation of a condition similar to Euclid's parallel axiom for subsets of finite sets. The background material to the theory of parallelisms is introduced and the author then describes the links this theory has with other topics from the whole range of combinatorial theory and permutation groups. These include network flows, perfect codes, Latin squares, block designs and multiply-transitive permutation groups, and long and detailed appendices are provided to serve as introductions to these various subjects. Many of the results are published for the first time.

Combinatorial designs and configurations --- Permutation groups --- Parallels (Geometry) --- Combinatorial designs and configurations. --- Parallels (Geometry). --- Permutation groups. --- Mathematics

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Group theory --- Permutation groups. --- Groupes de permutations --- Permutation groups --- 512.542.7 --- Statistics --- -Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Mathematics --- Econometrics --- Problems, exercises, etc --- -Permutation groups --- 512.542.7 Permutation groups --- -512.542.7 Permutation groups --- Statistical analysis --- Substitution groups

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Representation theory plays an important role in algebra, and in this book Manz and Wolf concentrate on that part of the theory which relates to solvable groups. The authors begin by studying modules over finite fields, which arise naturally as chief factors of solvable groups. The information obtained can then be applied to infinite modules, and in particular to character theory (ordinary and Brauer) of solvable groups. The authors include proofs of Brauer's height zero conjecture and the Alperin-McKay conjecture for solvable groups. Gluck's permutation lemma and Huppert's classification of solvable two-transive permutation groups, which are essentially results about finite modules of finite groups, play important roles in the applications and a new proof is given of the latter. Researchers into group theory, representation theory, or both, will find that this book has much to offer.

Solvable groups. --- Representations of groups. --- Permutation groups. --- Substitution groups --- Group theory --- Group representation (Mathematics) --- Groups, Representation theory of --- Soluble groups --- Solvable groups --- Representations of groups --- Permutation groups