Introductory description

The concept of a group is defined abstractly (as set with an associative binary operation, a neutral element, and a unary operation of inversion) but is better understood through concrete examples, for instance:
-permutation groups
-matrix groups
-groups defined by generators and relations. All these concrete forms can be investigated with computers. In this module we will study groups by
-finding matrix groups to represent them
-using matrix arithmetic to uncover new properties. In particular, we will study the irreducible characters of a group and the square table of complex numbers they define. Character tables have a tightly-constrained structure and contain a great deal of information about a group in condensed form. The emphasis of this module will be on the interplay of theory with calculation and examples.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Groups, subgroups. Examples (cyclic, dihedral, symmetric, alternating, matrix). Representations, equivalence of representations; the group ring, modules over the group ring. Sums of representations. Averaging and complements. Maschke's Theorem. Simple representations. Schur's Lemma. Traces and the character of a representation; Class functions.

Characters of fixed points. Characters of the representation Hom(V,W).
Hermitian inner products. Inner product of representations. Orthogonality of simple characters.
Simple characters give an orthonormal basis of class functions. The character table. Row and column orthogonality.

Inflated representations. Kernels and normal subgroups from characters. Abelian groups. Abelianizations and the centre from the character table.
Tensor products. Symmetric and alternating squares. Restricted and induced representations and their characters. Frobenius reciprocity.
Integrality. The dimensions of simples divide the group order. Integral values of characters. Burnside's Theorem.

Learning outcomes

By the end of the module, students should be able to:

• understand matrix and linear representations of groups and their associated modules,
• compute representations and character tables of groups, and
• know the statements and understand the proofs of theorems about groups and representations covered in this module.

Indicative reading list

We will work through printed notes written by the lecturer.
A nice book that we shall not use is:
G James & M Liebeck, Representations and Characters of Groups, Cambridge University Press, 1993. Second edition, 2001. (IBSN: 052100392X).

Subject specific skills

Problem solving, calculation and experiment leading to clarity and proof. The importance of systematic and symmetrical organization. The complementarity of example and abstraction.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.