Direct and semidirect products of groups. Dihedral groups and groups of matrices. Minimal normal subgroups and characteristically simple groups. Solvable groups. Actions and their applications. Primitive and imprimitive groups. Normal regular subgroups in a k-transitive group. A_n and PSL(n,F) as simple groups. Free groups and their isomorphisms. Presentations and relations. Descending central series and nilpotency. Finite nilpotent groups. Dedekind theorem.
The notes of prof. Carlo Casolo are the main reference and inspire my lessons. Anyway I follow a different path for some topics. In particular Kurzweil, Stellmacher, The theory of finite groups for the semidirect products; Robinson, A course in the theory of groups groups of matrices and free groups.
Other books are a useful complement. I recommand Milne, Group theory and Bogopolski, Introduction to group theory.
Some exercises and some seminars will come from the Notes of Prof. M. Garonzi.
Learning Objectives
To get acquainted with the theory of groups focussing on the finite case but having a frequent comparison with the infinite case.
Prerequisites
The content of the courses Algebra I and II.
Basic principal definitions and facts on groups, rings and fields.
Teaching Methods
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Training sessions: The students, in collaboration with the teacher, present exercises or small seminars on topic suggested by the teacher or by themselves.
The training sessions aim to:
-- help the students develop communication skills and apply the theoretical knowledge;
-- encourage independent learning;
-- let the students be an active part of the course developing, bearing up their interests.
Final oral examination: An exercise similar to those discussed in the training sessions is proposed. The tests are conceived to assess the ability of the in solving problems with correctness and formal correctness using a rigorous language.
A number of questions are then posed to evaluate the degree of understanding of the theory presented in the course. In the assessment, attention is paid to communication skills and the use of a proper mathematical language.
Course program
Groups and Lagrange Theorem. Products of subsets. Omomorphisms, normal subgroups and quotients. The three theorems of homomorphism. Characteristic subgroups. Direct products. Group of the automorphisms of a direct product. Group of the automorphisms of a cyclic group. Aut(G), Inn(G) and Out(G). Centralizers and normalizers. Normal clousure and Core of a subgroup. Semidirect products and Dihedral groups. Affine linear transformations of a field. Semidirect products between C_n and C_m. Every group generated by two involutions is dihedral. The center of a dihedral group. Groups of matrices and Singer cycles. General and special linear groups; general projective and special projective linear groups. Triangular T and unitriangular U matrices. U as the p-Sylow of GL(n,p^f), T as its normalizer. Omega series. Normal minimali subgroups and characteristically simple groups. Principal series for S_n. Solvable groups. Commutators and their computation. Lemma of the three subgroups. Derived series. Actions of a group G on a set or on a group. Kernel and image, faithfulness, orbits, stabilizers, Fix set. Actions in classical contexts and their applications. Burnside Lemma. If p is the minimum prime dividing |G|, then every subgroup of G of index p is normal. Classifications of the groups of order pq with p>q. Properties of the Sylow subgroups of a group. Every subgroup of G containing the normalizer of a Sylow is selfnormalizing. Frattini argument. The subgroup O_p(G). Equivalent actions. Two faithful and not equivalent action of S_5 on a set of 6 objects. Transitive actions. Multilply transitive groups. Primitive and imprimitive groups. Every normal subgroup of a primitive group is transitive. Every transitive and abelian permutation group is regular. Classification of the normal regular subgroups of a k-transitive group. Every normal abelian of a primitive group is regular. The only k-transitive group, with k greater than 3, containing a normal abelian not trivial subgroup is S_4. Primitive groups containing an m-cycle and recognition of subgroups of S_n. Simple groups. A_n and the special projective finite linear groups. SL(n,F) is perfect. PSL(n,F) is simple for n>2 and for n=2 with |F|>3. PSL(2,2) is isomorphic to S_3; PSL(2,3) is isomorphic to A_4. PSL(2,4) is isomorphic to PSL(2,5) and to A_5. Applications of the actions of groups. Actions of direct products of permutation groups on the so called preference profiles in social choice theory.Free groups and their construction. Isomorphisms of free groups. Normal form. The center of a free group. The free group of rank 2: its free subgroups of rank r for every r. Every group is a quotient of a free group. Presentations and relations. Free groups in “nature”. The ping-pong lemma. Nilpotent groups Descending central series and nilpotency. Central series. Recorsive formula for the terms of the descending central series. If G is nilpotent then its descending series is central. Solvable and nilpotent groups comparison. The ith center of a group. Link between a central series and the descending and ascending series. The class of nilpotent groups is closed by subgroups, quotients and direct product; it is not closed by extensions. In a nilpotent group every normal subgroup intersects not trivially the center. A maximal subgroup of a finite nilpotent group has prime index. Subnormal groups. Every subgroup of a nilpotent group is subnormal. Equivalent conditions for nilpotency for finite groups.The quaternion group Q_8 and the Dedekind theorem. Possible further topics: Frobenius groups. Schur-Zassenhaus theorem.