M4261

M4261 Finite groups

Aims: There are two aims to this unit. The first is to provide an introduction into basic algebraic and combinatorial methods whose applications go far beyond the theory of finite groups.
The second aim is to provide techniques that will enable us to classify certain finite groups and classes of finite groups. In particular, we will focus on more recent results that have been obtained for finite soluble groups, i.e. groups whose building blocks are cyclic groups of prime order.

Syllabus: A selection from the following topics: Homomorphisms and isomorphisms. Sylow theorems. Finite abelian groups and their complete classification. Special subgroups and subgroup series: commutator series, central series, chief series, composition series, p-series. Groups of prime-power order, nilpotent groups. Supersoluble groups. Sylow-type theorems. Covering-avoidance property. Formations of groups and generalisations; projectors. Fitting classes, injectors. Normal Fitting classes: Lockett sections and their Lausch groups. Unsolved problems in the theory of finite soluble groups.

Prerequisites: Some basic experience with numbers and/or structures that may have been obtained from units such as MTH1112 and/or MTH2122 / MTH3122.

References: Doerk, K. and Hawkes, T., Finite soluble groups, Walter de Gruyter, 1992.
Weinstein, M. (ed.), Between nilpotent and solvable, Polygonal Publishing House, 1982.
Gruenberg, K.W. and Roseblade, J.E., Group theory: essays for Philip Hall, Academic Press, 1984.
Zappa, G., Topics on finite solvable groups, Istituto nazionale di alta matematica Francesco Severi 1982.

Lecturer: Hans Lausch

For current students