Introduction to Group Theory

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.