The unit

• Taught by Dr Killian O'Brien + ...
• 2 hours lecture + 1 hour tutorial/lab per week
• Coursework problems (40%), Examination (60%)
• New(ish) unit: introduced in response to student requests for more pure mathematics content

Nature of the unit

• A thorough introduction to a substantial area of pure mathematics that has strong connections to areas of geometry, combinatorics, graph theory, ... .
• Definitely suited to students who like problem solving and the unit will develop your skills in this area.
• We will use the SageMath mathematics system to aid our investigations. You will also get an introduction to the Python programming language. (www.sagemath.org, cloud.sagemath.org, sagecell.sagemath.org)
• Excellent textbook: Abstract Algebra by Tom Judson

What you will get from the unit

• An appreciation for the many aspects of group theory and its connections to other areas of mathematics.
• Development of your problem solving and abstract thinking skills.
• Exposure to SageMath mathematical software, aimed at pure mathematics teaching and research.
• SageMath uses the Python programming language which is widely used in many areas of computing.

What is a group?

A set of mathematical objects with a mathematically meaningful operation applied amongst them, as long as it is well behaved, it will be a group. It's a very broad concept and present in many areas of mathematics.

More formally

A set \(G\) with a binary operation \(\star\) satisfying

• \(G\) is a closed system under \(\star\), i.e. \(x \star y \in G\).
• \(\star\) is associative on \(G\), i.e. \(x \star ( y \star z) = (x \star y) \star z\).
• \(G\) contains an identity element, \(e\), for \(\star\), i.e. \(x \star e = e \star x = x\).
• \(G\) contains inverse elements for \(\star\), i.e. \(x \star x^{-1} = e\).

Syllabus topics

Examples of groups:

• Symmetry groups of two and three-dimensional objects,
• the dihedral groups \(D_n\),
• rotational symmetry groups of three-dimensional polyhedra.
• Permutation groups, the Symmetric groups \(S_n\) and the Alternating groups \(A_n\).
• Number based groups under arithmetic operations,
• the cyclic groups \(\mathbb{Z}_n\).
• Groups of matrices.
• ...

Visualising the elements of \(\Gamma^+(T)\)

\(\Gamma^+(T)\) is the set of rotational symmetries of the tetrahedron.

The Symmetric group, \(S_4\), on four objects

\(S_4\) is the group of all rearrangments, or permutations, of four objects.

Syllabus topics

Initial group theory:

Various concept definitions and examples, including: elements, orders, Abelian groups, subgroups, generators and cyclic/non-cyclic. The natural mappings between groups, homomorphisms and isomorphisms. Examples of isomorphic pairs and non-isomorphic pairs. Cayley's theorem: Every group isomorphic to a group of permutations.

Syllabus topics

Classification problems:

What are the grand enterprises of group theory? What classification problems can be posed?

Lagrange's theorem:

Restricting the possibilities for subgroup orders. Equivalence relations, equivalence classes, cosets. Normal groups and quotient groups.

Lagrange's theorem

A page from Groups and Symmetry by M. A. Armstrong.

Syllabus topics

Group presentations:

How to systematically describe groups in a computable way. Group presentations, generators and relations, presentation matrices. The isomorphism decision problem based on matrices.

The classification of finitely presented Abelian groups:

A matrix reduction algorithm to decide the isomorphism problem amongst finitely presented Abelian groups. The canonical form of finitely presented Abelian group as a direct sum of cyclic groups.

Syllabus topics

Classification of groups of low order:

What about non-Abelian groups? Why we can't solve using matrix reduction? Investigation of groups of low order and enumeration and classification of all groups up to some suitable order.

Sylow's theorems:

Discussion of the converse to Lagrange's theorem. Group actions, orbits, stabilizers. Self-action by conjugation. Sylow's theorems.

Wider interest material / applications

The unit could contain interesting general material on the following topics/applications.

The classification of finite simple groups

The grand project. Status of the proof. Some history and biographical details of the completion of the project. The families in the classification. The sporadic groups. The Monster group and Monstrous Moonshine.

Wider interest material / applications

The Monster group

• A group, \(M\), with approx. \(8 \times 10^{53}\) elements, that is simple, i.e. it has no normal subgroups.
• \(M\) is (isomorphic to) a group of rotations of 196883-dimensional space.
• \(M\) is (isomorphic to) a group of matrices generated by two particular binary \(196882 \times 196882\) matrices.

Wider interst material / applications

Algorithmic problems

The word problem. Computability. Alan Turing.

Wider interst material / applications

Combinatorial enumeration and geometric classification problems

Counting number of distinguishable colourings of geometric objects. Classifying the symmetry types of two-dimensional wallpaper patterns. Classifying two and three-dimensional crystal structures (lattices).

The are exactly 17 different types of wallpaper, up to symmetry.

Wider interst material / applications

Galois theory

• Life of Galois (1811 - 1832).
• Galois theory.
• Formulas for roots of polynomials.
• Construction problems with ruler and compass.

SageMath

G=DihedralGroup(12)
# G is the group of rotational and
# reflectional symmetries of the
# regular 12-sided polygon

HH = G.subgroups()
# HH is the set of all subgroups of G

# The following for loop prints the
# Cayley graph of each subgroup of G
for H in HH:
    show(H.cayley_graph())

The above code displays the Cayley Graph of every subgroup of \(D_{12}\). See the results from SageMath.