Groups and Symmetry
Semester 1
Credits 10
Prerequisites: | PMA212 Linear Mathematics II, PMA216 Algebra |
Corequisites: | None |
Cannot be taken with: | None |
Description
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, and in combination with Sylow's theorems, this gives a powerful way of constructing groups.
Aims
Outline Syllabus
Module Format
Lectures | 20 | Tutorials | 0 | Practicals | 0 |
Recommended Books (A=core text, B= secondary text, C=background reading)
Assessment
One formal 2.5 hour exam. Format: 4 out of 5 questions.
More details from Dr N.P. Strickland (Room J10, Telephone 23852)
Full Syllabus
1. Symmetry groups in R^{n} (1 lecture)
Orthogonal and special orthogonal groups, O_{n} and SO_{n}. [O_{n}:SO_{n}] = 2. O_{n} consists of reflections and rotations. Basic properties of reflections and rotations. Symmetry and direct symmetry groups, Symm(X) and Dir(X).
2. Linear symmetry groups of R^{2} (1 lecture)
Dihedral and cyclic groups as symmetry and direct symmetry groups of a regular n-gon. Basic properties of D_{n}. Finite subgroups of O_{2} are dihedral or cyclic.
3. Wallpaper patterns I: the isometry group of R^{2}. (2 lectures)
Group of isometries and groups of translations of R^{n}, I_{n} and T_{n}. T_{n} @ (R^{n},+). Examples in R^{2}: reflections, rotations, glides. An isometry of R^{n} fixing O is linear. I_{2} consists of translations, rotations, reflections and glides. An isometry of R^{n} is determined by its values on x_{i}, i = 0,...,n if (x_{1}-x_{0}, ... , x_{n}-x_{0}) are linearly independent.
4. Semidirect products. (2 lectures).
Semidirect product, definition and characterisation. Examples to include D_{n} = C_{n} \rtimes áSñ and I_{n} = T_{n}\rtimesO_{n}. D_{n} as a semidirect product. Generalisation of determinant function to isometries of the plane. Direct products are semidirect products. Construction of a semidirect product from an action through group homomorphisms.
5. Wallpaper patterns II: examples. (1 lecture)
Isometry group, I(X). Isometry groups of wallpaper patterns. Point groups of wallpaper patterns.
6. Linear groups of symmetries of R^{3}.
Preliminaries.(1 lecture)
Review of group actions on sets equivalence between G actions on X and homomorphisms G® S(X), the symmetric group of X; D_{n} as a subgroup of S_{n}, for n ³ 3.
Reflections in a hyperplane. Every rotation in SO_{3} has an axis.
The tetrahedron.(1/2 lecture)
Faithful action on vertices. Symm(Tet) @ S_{4}. If G £ O_{n} then [G:GÇSO_{n}] £ 2. Dir(Tet) @ A_{4}.
The cube. (1/2 lecture)
Faithful action of Dir(Cube) on long diagonals. Description of 24 rotations in Dir(Cube). Dir(Cube) @ S_{4}. G £ O_{3} and -1 Î G implies that G @ GÇSO_{3}×{±1}. Symm(Cube) @ S_{4}×{±1}.
The Dodecahedron. (1 lecture)
Construction via tents on a cube. Faithful action of Dir(Dodec) on 5 inscribed cubes. Description of 60 rotations in Dir(Dodec). Dir(D) @ A_{5}. Symm(Dodec ) @ A_{5}×{±1}. A_{5} only proper, nontrivial normal subgroup of S_{5}.
Finite rotation groups in R^{3} (2 lectures)
Platonic solids. Icosahedron as dual of dodecahedron, octahedron as dual of the cube. Duals have same symmetry groups. Poles on the 2-sphere. Stabilisers of poles are cyclic. Finite subgroups of SO_{3} are cyclic, dihedral or the direct symmetry group of a Platonic solid.
7. The Sylow theorems. (2 lectures)
Finite p-groups and p-subgroups. Statement of Sylow theorems. Sylow p-subgroups. Examples of Sylow theorems (orders 6,12 and 20; Sylow subgroups of S_{4}). Any group of order pq (with p ¹ q) is a semidirect product. Proof of Sylow theorems. |H||K| = |HK||HÇK|, for subgroups H and K.
8. Automorphisms. (1 1/2 lectures)
Automorphism group of a group. Examples. Reminder of subgroups of a cyclic group. Aut(C_{r}) as the group of units of \mathbbZ_{r} and more generally automorphisms of elementary abelian groups.
9. Groups of small order. (2 1/2 lectures)
Examples involving combined use of Sylow's theorems and semidirect products. Classification of groups of order 12. Quaternion group of order 12, Q_{12}. The classification of groups of order £ 15.
10. Finite simple groups. (2 lectures) Simple groups. The classification. SL_{2}(Z/p). Subgroups of order p and p-1. The centre of SL_{2}(Z/p). The group PSL_{2}(Z/p). The simplicity of PSL_{2}(Z/p) for p ³ 5; proof for p = 7.