Introduction to Group Theory

The present book is the outcome of a one-semester lecture course which the author has given frequently during the last three decades. The course has been gradually modified over the years in accordance with changing outlook and with the steadily increasing sophistication of the audience, third- and fourth-year honours classes in several universities in Australia and in Canada. Out of the conviction that no branch of Mathematics can be mastered by memorizing facts and methods I have tried from the beginning to make the subject interesting for the reader. Clearly one cannot hope to please everybody in this respect. I have sought, how­ ever, to attract the reader's interest by including a number of dis­ cussions and examples which either have interested me on occasion or resulted from questions of students; some sections and examples have been taken from work done by now eminent mathematicians in an early period of their career, assuming that such selections will appeal to younger readers. So J hope that this book will be found to be a reasonably modern, although not conventional, text proposing an amount of material most of which can be dealt with in a half-year's lecture course. After studying the book the reader should be able to tackle those problems in group theory which are scattered in the problem sections of the American Mathematical Monthly and other similar periodicals.

I Definition of a Group and Examples.- §1 The abstract group and the notion of group isomorphism.- a. Sets and mappings.- b. Algebraic systems.- c. Semigroups.- d. Groups.- e. Isomorphism.- f. Cyclic groups 7.- Examples and exercises.- §2 Groups of mappings. Permutations. Cayley’s theorem.- a. Composition of mappings.- b. Permutations.- c. Cycles.- d. Transpositions.- e. Subgroups.- f. Cayley’s theorem. The group table.- Examples and exercises.- §3 Arithmetical groups.- a. Facts of number theory.- b. Residue classes modulo m. Euler’s function.- c. The unit group of a ring.- d. Matrix residue class groups (mod m).- e. The case m = 2.- Examples and exercises.- §4 Geometrical Groups.- a. Rotations and reflexions.- b. The dihedral groups.- c. Rotations and reflexions in space.- d. The polyhedral groups.- e. The groups of the octahedron and of the tetrahedron.- Examples and exercises.- II Subsets, Subgroups, Homomorphisms.- §1 The algebra of subsets in a group.- a.–c. Product and inverse of subsets.- d. Subgroup generated by a subset.- e. Two disjoint subsets covering a group.- Examples and exercises: Frattini subgroup.- §2 A subgroup and its cosets. Lagrange’s theorem.- a. Cosets. Lagrange’s theorem.- b. Index theorems.- c. Poincaré’s theorem.- d. Finite cyclic groups.- Examples and exercises: Multiplicative group of a finite field. The icosahedral group. Frattini subgroup. Abelian groups.- §3 Homomorphisms, normal subgroups and factor groups.- a. Homomorphism, epimorphism, monomorphism.- b. Kernel.- c. Natural homomorphism and factor group.- d. Canonical product.- Examples and exercises.- §4 Transformation. Conjugate elements. Invariant subsets.- a. Conjugacy.- b. Invariance..- c. The classes.- d. The normalizer.- e. Generalization of Cayley’s theorem.- Examples and exercises: Transformation in permutation groups. Geometry in a group.- §5 Correspondence theorems. Direct products.- a.–b. Theorems.- c. The internal direct product.- d. Generalization to more than two factors.- e. The external direct product.- f. The restricted direct product.- Examples and exercises.- §6 Double cosets and double transversals.- a. A counting formula.- b. Double cosets.- c. Double transversals.- d. A group of double cosets.- Examples and exercises.- III Automorphisms and Endomorphisms.- §1 Groups of automorphisms. Characteristic subgroups.- a. Generalities.- b. Inner automorphisms.- c. Characteristic subgroups.- d. Characteristically simple groups.- e. ?-invariance.- Examples and exercises: Automorphism groups of some special groups. Simplicity of the alternating groups. The kernel or nucleus of a group. The Frattini subgroup, a characteristic subgroup.- §2 The holomorph of a finite group. Complete groups.- a. Definition of the holomorph.- b.–c. The holomorph as a permutation group.- d. Is the holomorph minimal?.- e. Complete groups.- Examples and exercises: Completeness of the symmetric groups. Equicentralizer systems in groups.- §3 Group extensions.- a. The semi-direct product.- b. The external semi-direct product.- c. Are there other solutions to the extension problem ?.- d.–e. Construction of a normal extension.- Examples and exercises: Complement. Splitting extension.- §4 A problem of Burnside: Groups with outer automorphisms leaving the classes invariant.- a. Preliminaries on the groups L2n.- b. The groups L4n and L8n.- c. Automorphism associated with a subgroup of index 2.- d. Proof of the theorem concerning L8n.- Examples and exercises.- §5 Endomorphisms and operators.- a. Endomorphisms.- b. The endomorphism ring of an abelian group.- c. Operator domain of a group. Fully invariant subgroups.- Examples and exercises.- IV Finite Series of Subgroups.- §1 The fundamental concepts of lattice theory.- a. Partially ordered sets.- b. Lattices.- c. Partially ordered set and lattice.- d. Modular lattices and distributive lattices.- Examples and exercises.- §2 Lattices of subgroups.- a. The lattice of all subgroups of a group.- b. The lattice of admissible subgroups.- c. The lattice of normal subgroups.- d. The Lemma of Zassenhaus.- Examples and exercises.- §3 The theory of O. Schreier.- a. Chains and series of subgroups.- b. Refinement of series. Schreier’s theorem.- c. The theorem of Jordan and Hölder.- d. Applications.- e. Solvable groups.- Examples and exercises: Maximum normal subgroups.- §4 Central chains and series.- a. The ascending central chain.- b. The upper central chain.- c. Nilpotent groups.- d. Mixed commutator subgroups.- e. The lower central chain.- Examples and exercises.- V Finite Groups and Prime Numbers.- §1 Permutation groups.- a. Action of a group on a set.- b. Transitivity.- c. Stabilizers.- d. Application.- e. Multiple transitivity. Imprimitivity.- f. Sylow’s first theorem.- Examples and exercises: Bertrand’s theorem.- §2 Sylow’s theorems.- a. Cauchy’s theorem.- b. The class equation.- c. Theorem 1.- d. Theorem 2.- e. Theorem 3.- f. Theorem 4.- Examples and exercises: Landau’s theorem. Theorems on finite p-groups. The groups of order pq.- §3 Finite nilpotent groups.- a. A direct product of p-groups.- b. Necessity.- c. Maximal subgroups. Schmidt’s theorem.- d. The Frattini subgroup.- Examples and exercises: Gaschütz’s theorem.- §4 The structure of finite abelian groups.- a. Existence of a basis.- b. Uniqueness (invariance) of the orders of the basis elements.- c. Application to the construction of abelian groups.- Examples and exercises: The residue class group ? m.- Appendix Hints or Solutions to Some of the Exercise Problems.- Author index.