| Credits: 20 | Convenor: Dr. W. Wheeler | Semester: 1 |
| Prerequisites: | essential: MC242 | |
| Assessment: | Coursework: 10% | Three hour exam: 90% |
| Lectures: | 36 | Classes: | 12 |
| Tutorials: | none | Private Study: | 102 |
| Labs: | none | Seminars: | none |
| Project: | none | Other: | none |
| Total: | 150 |
One of the module's major goals is to develop enough theory to be able to discuss in a worthwhile fashion the classification of the finite simple groups. This classification ranks as one of the major intellectual achievements of all time. Its proof runs to somewhere between 10,000 and 15,000 journal pages, spread across some 500 separate articles by more than 100 mathematicians, almost all written between 1950 and the early 1980's. A revision is currently underway, but even this is expected to run to more than 5,000 pages.
The module presents an outline description of the classification, explains its significance, and gives a hint of the complexity of its proof.
DEFINITIONS AND EXAMPLES Definition of a group, infinite/finite/abelian/cyclic groups, multiplication tables, order of a finite group, the cyclic groups Cn, groups of matrices under addition and multiplication, groups of symmetries, the dihedral groups Dn, direct products.
MAPS AND RELATIONS ON SETS Maps, surjections, injections, bijections, compositions, identity maps, inverse of a map, composition of injections/surjections/bijections, permutations, the symmetric groups Sn, cycle notation for and multiplication of permutations, group isomorphism, relations, equivalence relations, partitions, inclusion maps, quotient maps, writing a map as the composition of a quotient map, a bijection and an inclusion map.
ELEMENTARY CONSEQUENCES OF THE DEFINITIONS Uniqueness of identity/inverse, solving equations, inverse of a product and of an inverse, superfluity of brackets, definition and manipulation of powers, order of an element, order of a permutation, equality of powers, groups of order 3 and 4.
FREE GROUPS Definition of a free group on a set X, verification of group axioms, rank of a free group.
SUBGROUPS Definition, subgroup criteria, examples of subgroups, generators and generating subgroups, cyclic subgroups and their order, generators for Sn, subgroups of cyclic groups, centralizers of elements and of subsets, subgroup of free group comprising words of even length.
PRODUCTS OF SUBSETS The notation AB where A,B are both subsets of a group G, subgroup criteria using such notation.
COSETS AND LAGRANGE'S THEOREM Cosets as special case of products of subsets, examples, cosets as equivalence classes, bijections between cosets, Lagrange's theorem, index of subgroups, order of an element divides order of the group, groups of prime order are cyclic, groups of order 6 and 8, finite subgroups of the non-zero complex numbers under multiplication.
NORMAL SUBGROUPS AND QUOTIENT GROUPS Conjugates of elements and of subsets, normal subgroups and equivalent criteria, examples of normal subgroups, intersections of normal subgroups, product of a subgroup and a normal subgroup, quotient groups: definition and verification of group axioms, examples, subgroups of quotient groups, centre of a group.
HOMOMORPHISMS Definition, elementary properties of homomorphisms, examples, the signature map on permutations, the determinant map on matrices, homorphisms from free groups, homomorphisms onto quotient groups, isomorphism as an equivalence relation, kernels, the alternating groups An, the homomorphism theorem, the isomorphism theorems, automorphism groups and the conjugation map.
PRESENTATIONS AND COSET ENUMERATION
Definition of 
as a quotient group of the free group F(X),
defining relations for a group, examples,
coset enumeration, the problem of determining finiteness.
G-SETS, ACTIONS AND THE ORBIT-STABILIZER THEOREM G-sets/actions: definitions and examples, G-sets as homomorphisms to symmetric groups, kernel of an action, faithful/transitive actions, multiplication/conjugation actions, stabilizers, orbits, orbit-stabilizer theorem, more on conjugacy classes, finite p-groups have non-trivial centres, finding normal subgroups as union of conjugacy classes, groups of order p2.
SYLOW THEOREMS Sylow p-subgroups of finite groups, the Sylow theorems, elementary properties of Sylow subgroups, normalizer of a Sylow subgroup has a unique Sylow subgroup, characteristic subgroups and their relationship to normal subgroups and to unique Sylow subgroups, applications: every finite group whose order is divisible by prime p has an element of order p, existence of normal subgroups in groups of various composite orders.
JORDAN-HöLDER THEOREM Subnormal/normal/isomorphic/refinable/composition/chief series, examples, Jordan-Hölder theorem.
COMPOSITION FACTORS AND CHIEF FACTORS
Definition, simple groups, characteristically simple groups, simplicity of the
alternating groups 
, maximal/minimal normal subgroups, constructing
composition and chief series.
SOLUBLE AND NILPOTENT GROUPS Soluble groups, commutator subgroups, examples, central series and nilpotent groups, p-groups are nilpotent, maximal subgroups of nilpotent groups are normal, solubility of groups of order less than 60.
THE CLASSIFICATION OF THE FINITE SIMPLE GROUPS Discussion of the classification of the finite simple groups.
J. F. Humphreys, A Course in Group Theory, Oxford University Press.
J. A. Gallian, Contemporary Abstract Algebra, DC Heath.
D. L. Johnson, Presentation of Groups, Cambridge University Press.
C. R. Jordan and D. A. Jordan, Groups, Edward Arnold.
I. D. MacDonald, Theory of Groups, Oxford University Press.
J. S. Rose, A Course in Group Theory, Cambridge University Pres.
J.J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag.