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Permutation groups. (English) Zbl 0951.20001

Graduate Texts in Mathematics. 163. New York, NY: Springer-Verlag. xii, 346 p. (1996).
Since the publication in the sixties of the books by H. Wielandt and D. S. Passman on (finite) permutation groups there has been a fundamental change in the theory of permutation groups by the classification of finite simple groups obtained about 1980. Many longtime open problems have been solved by the newly available methods based on the O’Nan-Scott theorem, using the classification.
The present book by Dixon and Mortimer takes account of this important fact. It can be used for lectures addressing graduate students, for self-study of the subject and for research as a first reference to fundamental facts and most important methods.
Emphasis is given to the theory of finite permutation groups, but fundamental concepts and results are developed and presented without finiteness assumptions whenever possible. The last chapter of the book deals with some recent topics in the theory of infinite permutation groups.
A sketch of the contents is as follows: In Chapter 1 the concepts of transitivity and primitivity for group actions and permutation groups are introduced and developed. Chapter 2 deals with actions induced by combinatorial settings; wreath products in both natural actions (imprimitive and primitive) are dealt with and affine resp. projective linear groups in their natural actions are discussed.
Chapter 3 deals with some selected topics: orbits of point stabilizers resp. orbitals, minimal degree, bases, Frobenius groups and other groups having a regular normal subgroup. Also a short introduction to permutation group algorithms is given which is useful for application of computer algebra software like GAP, MAGMA, MAPLE or MATHEMATICA to problems in permutation groups.
Chapter 4 and 5 deal with the theory of finite primitive permutation groups. In particular, a proof of a version of the O’Nan-Scott theorem on the structure of finite primitive permutation groups is given. This theorem allows the reduction of many problems to the study of primitive almost simple permutation groups and to the investigation of irreducible modular representations over fields of prime number order. Some applications are discussed, in particular the proof of Sims’ conjecture by Cameron, Praeger, Saxl and Seitz.
In Chapter 5 bounds for the order, minimal base size and minimal degree of finite primitive or even doubly-transitive permutation groups are derived (results due to C. Jordan, Bochert, Wielandt, Babai, Pyber, Praeger and Saxl). For this the methods are combinatorial and number theoretic; Liebeck’s theorem (1984) on the minimal degree of primitive groups which uses the O’Nan-Scott theorem and the classification is quoted without proof.
In Chapter 6 the Mathieu groups are dealt with on the basis of their associated Steiner systems. Chapter 7 is devoted to the general theory of multiply transitive permutation groups, in particular to the classification of finite doubly-transitive permutation groups.
Chapter 8 contains a discussion of symmetric groups of arbitrary degree with respect to their normal subgroups and to their subgroups which are maximal or have small index. In Chapter 9 some interesting classes of infinite permutation groups are studied: groups acting on infinite trees, highly transitive free groups, homogeneous groups and the automorphism group of the universal graph.
Two appendices are attached: Appendix A gives a very short survey of the results of the classification of finite simple groups; Appendix B gives a list of primitive permutation groups of degree less then \(1000\), originally published in the authors’ paper [Math. Proc. Camb. Philos. Soc. 103, No. 2, 213-238 (1988; Zbl 0646.20003)]. The list of references contains more than 300 items of the current literature.

MSC:

20Bxx Permutation groups
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20B05 General theory for finite permutation groups
20B07 General theory for infinite permutation groups
20B10 Characterization theorems for permutation groups
20B15 Primitive groups
20B20 Multiply transitive finite groups
20B22 Multiply transitive infinite groups
20B27 Infinite automorphism groups
20B30 Symmetric groups
20B35 Subgroups of symmetric groups
20B40 Computational methods (permutation groups) (MSC2010)

Citations:

Zbl 0646.20003

Software:

Mathematica; Maple; GAP; Magma
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