1.  Introduction:  This fourth year course is a thorough introduction to Finite Groups. Group theory is a large topic which interconnects with many branches of pure and applied mathematics. The unit is a good accompaniment to other pure mathematics units such as Representation Theory, Ring Theory and Algebraic Number Theory. (It is good advice where practicable to do these units concurrently or after the group theory course.)

2.  Hours, Credits and Assessment:  The course is a 20 UCU unit of 33 lectures and some additional hours for discussion in class. Assessment is by course work (20%) through assessed homework and examination (80%).

3.  Overview:  Group Theory has two main roots, one in geometry where groups of geometrical transformations were studied, the other in algebra and the theory of equations where groups of substitutions of variables (i.e. permutations) in polynomial functions were analysed. The revolutionary work of Galois (1820's) about the solvability of polynomial equations, for instance, made it necessary to study such groups of substitutions. Finite group theory evolved to a great extent from this second root. Abstract groups began to emerge with Jordan's seminal Traité des substitutions et des equations algébriques (1870) while the definition of abstract groups in general appears to be due to Weber (1882).

The course starts with a review of elementary facts such as the correspondence and isomorphism theorems, composition series and composition factors. The theorem of Jordan and Hölder will be proved which shows that a finite group is constructed in some fashion from simple groups.

The idea that a group acts on a set is fundamental. Therefore group actions will be studied in great detail. For finite groups the orbit stabilizer theorem, a relatively easy result on group actions, plays a central role and many theorems appear as a  consequence of it. One instance is the theorem which determines the number of orbits of a permutation group which is used for pattern counting more generally. Other applications include the Class Equation which leads to an elementary introduction to groups whose order is a power of a prime. Group actions are also fundamental for the study of the Platonic bodies.

Sylow's Theorem is an early high point of finite group theory. It says that a finite group of order x. p^n with p a prime not dividing x has a subgroup of order p^n. It also says that any two such subgroups are conjugate and that their number is congruent to 1 modulo p. Several proofs of this theorem will be given, various strengthenings will be proved and a variety of numerical non-simplicity will be derived.

Commutative groups which are generated by a finite number of elements can be analysed completely. This theorem has many applications outside group theory, including in matrix theory, number theory and topology.

4.  Recommended literature and references:  While there is no single text book for the course there are a great number of books on group theory. The following list begins with some more basic texts which are useful to look up elementary facts about groups:

(1) Herstein: Topics in Algebra. (Useful for the beginner, does not cover all you need.)

(2) PM Cohn: Algebra. (Useful for the beginner, does not cover all you need.)

(3) J Rotman: Introduction to Group Theory, Springer Verlag. (Contains all of what you need and much more.)

(4) J Rose: A Course on Group Theory, CUP. (Contains a lot of exercises and examples.)

(5) M Hall: The Theory of Groups, Macmillan. (A thorough treatment, not in print anymore.)

(6) Burnside: Group Theory. (A classic text, not always easy to read.)

Not all of these books are still in print but all should be in the library.

1. Factor groups, the correspondence and isomorphism theorems. Composition series, composition factors and the Jordan-Hölder theorem. Solvable and simple groups. (6 lectures)

2. Permutation groups, the sign function and definition of the alternating group. Review of group actions and applications: action on cosets, conjugation. Simplicity of alternating groups. Orbit stabilizer theorem, orbit counting theorem. Examples: Symmetry groups of combinatorial structures such as Petersen graph. The rotation groups of the cube and dodecahedron. The Class equation for a finite group. (12 lectures)

3. The proof of the three Sylow theorems with various extensions. Counting Sylow subgroups with applications to non-existence of simple groups of certain orders. A5 is the only non-abelian simple group of order less than 100. (8 lectures)

4. Introduction to p-groups. The centre intersect any normal subgroup non-trivially, normalizers of subgroups, the Frattini argument. Nilpotency and the Frattini subgroup. Generators for a p-group. (4 lectures)

5. Finitely generated abelian groups. Row and column operations on integer matrices, Smith normal form of a an integer matrix. Fundamental theorem on finitely generated abelian groups. (3 lectures)

Finite sub-groups of GL(3,R) with applications to Platonic bodies, reflection groups and Coxeter groups.