## MAS346 Groups and Symmetry

Note: Information for future academic years is provisional. Timetable information and teaching staff are especially likely to change, but other details may also be altered, some courses may not run at all, and other courses may be added.

 Semester 2, 2022/23 10 Credits Lecturer: Dr Cristina Manolache Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

Groups arise naturally as collections of symmetries. Examples considered include symmetry groups of Platonic solids and of wallpaper patterns. Groups can also act as symmetries of other groups. These actions can be used to prove the Sylow theorems, which give important information about the subgroups of a given finite group, leading to a classification of groups of small order.

Prerequisites: MAS211 (Advanced Calculus and Linear Algebra); MAS220 (Algebra)
No other modules have this module as a prerequisite.

## Outline syllabus

• Orthogonal and special orthogonal symmetries of Rn
• Group actions, Sylow theorems, and simple groups
• Symmetry and direct symmetry groups
• Affine isometries
• Wallpaper groups
• Groups of symmetries of the platonic solids
• Groups of small order

## Aims

• To consolidate previous knowledge of group theory, symmetries and linear algebra
• To display and exemplify the ubiquity of groups as symmetries of physical and mathematical objects
• To introduce and illustrate the process of analysis of a finite group from its local structure

## Learning outcomes

On successful completion of this course, students should be able to:
• recognize and apply actions of groups on sets in geometric and abstract contexts
• carry out calculations with plane isometries
• analyse the isometry group of a wallpaper pattern
• give examples of simple groups and prove their simplicity
• understand the concept of the Sylow p-subgroup and the statement of the Sylow theorems
• apply Sylow theorems to rule out existence of simple groups of certain orders
• apply Sylow Theorems to analyse the structure of groups of small order
• determine the symmetry groups of geometric structures (including polygons, tetrahedron, cube, and dodecahedron).

## Teaching methods

Lectures, problem solving

20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination. All questions compulsory.

## Full syllabus

1. Symmetry groups in Rn
(3 lectures) Matrix groups GLn, SLn, inner product, orthogonal matrix preserves inner product. On, SOn, [On:SOn]=2, On consists of reflections and rotations. Basic properties of reflections and rotations.

2. Groups acting on sets and simple groups.
(4 lectures) Review of group actions on sets, Orbit-Stabilizer Theorem. Equivalence between G actions on X and homomorphisms G→ S(X), the symmetric group of X. Dn as a subgroup of Sn. Sylow p-subgroups. Sylow Theorems statement. Non-existence of simple groups of certain orders.
3. Symmetry Groups (2D)
(2 lectures) Symmetry and direct symmetry groups, Sym(X) and Dir(X). Dn and Cn as symmetry and direct symmetry groups of regular n-gon. Basic properties of Dn. Finite subgroups of O2 are cyclic or dihedral.
4. Affine isometries
(2 lectures) Group of isometries and groups of translations of \ren, I(\ren ) and T(\ren ). T(\ren ) ≅ (\ren ,+). Examples in \retwo: reflections, rotations, glides. An isometry of \ren fixing O is linear. I(\retwo) consists of translations, rotations, reflections and glides.
5. Wallpaper groups
(3 lectures) Isometry group, I(X). Isometry groups of wallpaper patterns. Point groups of wallpaper patterns.
6. Symmetry Groups (3D)
(3 lectures) Symmetry groups of the tetrahedron, the cube, and the dodecahedron. Finite subgroups of SO3 are cyclic, dihedral or the direct symmetry of a platonic solid (without proof).
7. Sylow Theorems: proof and further applications.
(3 lectures) The internal and external direct products of groups. Two criteria for a group to be isomorphic to a direct product of its subgroups. Proof of Sylow theorems. A description of groups of order pq where p and q are primes such that p < q.