Cheryl E. Praeger's Books

Permutation groups, their fundamental theory and applications are discussed in this introductory book. It focuses on those groups that are most useful for studying symmetric structures such as graphs, codes and designs. Modern treatments of the O'Nan–Scott theory are presented not only for primitive permutation groups but also for the larger families of quasiprimitive and innately transitive groups, including several classes of infinite permutation groups. Their precision is sharpened by the introduction of a cartesian decomposition concept. This facilitates reduction arguments for primitive groups analogous to those, using orbits and partitions, that reduce problems about general permutation groups to primitive groups. The results are particularly powerful for finite groups, where the finite simple group classification is invoked. Applications are given in algebra and combinatorics to group actions that preserve cartesian product structures. Students and researchers with an interest in mathematical symmetry will find the book enjoyable and useful.

This book presents a complete classification of the transitive permutation representations of rank at most five of the sporadic simple groups and their automorphism groups, together with a comprehensive study of the vertex-transitive graphs associated with these representations. Included is a list of all vertex-transitive, distance-regular graphs on which a sporadic almost simple group acts with rank at most five. In this list are some new, interesting distance-regular graphs of diameter two, which are not distance-transitive. For most of the representations a presentation of the sporadic group is given, with words in the given generators which generate a point stabiliser: this gives readers sufficient information to reconstruct and study the representations and graphs. Practical computational techniques appropriate for analysing finite vertex-transitive graphs are described carefully, making the book an excellent starting point for learning about groups and the graphs on which they act.

The main result describes completely the maximal factorizations of all the finite simple groups and their automorphism groups. As a consequence, a classification of the maximal subgroups of the finite alternating and symmetric groups is obtained.

Generating function techniques are used to study the probability that an element of a classical group defined over a finite field is separable, cyclic, semisimple or regular. The limits of these probabilities as the dimension tends to infinity are calculated in all cases, and exponential convergence to the limit is proved. These results complement and extend earlier results of the authors, G. E. Wall, and Guralnick & Lubeck.

Addresses the classical problem of determining finite primitive permutation groups G with a regular subgroup B.

Book's by Cheryl E. Praeger

Permutation Groups and Cartesian Decompositions

Permutation Groups and Cartesian Decompositions

Low Rank Representations and Graphs for Sporadic Groups

Low Rank Representations and Graphs for Sporadic Groups

The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups

The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups

A Generating Function Approach to the Enumeration of Matrices in Classical Groups over Finite Fields

A Generating Function Approach to the Enumeration of Matrices in Classical Groups over Finite Fields

Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster

Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster

Regular Subgroups of Primitive Permutation Groups

Regular Subgroups of Primitive Permutation Groups

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