Groups and Symmetry

Semester 1       

Credits 10

Prerequisites: PMA212 Linear Mathematics II, PMA216 Algebra
Corequisites: None
Cannot be taken with: None


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.


  1. To display and exemplify the ubiquity of groups as symmetries of physical and mathematical objects.
  2. To consolidate and unify previous knowledge of groups and symmetry.
  3. To introduce the idea of symmetry groups of geometric structures.
  4. To introduce and illustrate the process of analysis of a finite group from its local structure.

Outline Syllabus

  1. Orthogonal and special orthogonal symmetries of Rn and subsets.
  2. Isometries of Rn.
  3. Isometry groups of Rn and subsets.
  4. Wallpaper patterns.
  5. Classification of finite subgroups of O2 and SO3.
  6. Semidirect products.
  7. Automorphisms of a group.
  8. The Sylow theorems.
  9. Groups of small order.

Module Format
Lectures 20 Tutorials 0 Practicals0

Recommended Books (A=core text,  B= secondary text, C=background reading)

  1. [A/B] M.A.Armstrong ``Groups and Symmetry'' Springer-Verlag (1988).
  2. [A/B] E. Rees, ``Notes on geometry'', Springer-Verlag (1983).
  3. [B] R.C.Lyndon ``Groups and geometry'' LMS lecture notes 101 CUP (1985).
  4. [C] Grove and Benson, ``Finite Reflection Groups'', Springer-Verlag (1985).
  5. [C] M. Artin, ``Algebra'', Prentice Hall (1991).


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 Rn (1 lecture)

Orthogonal and special orthogonal groups, On and SOn. [On:SOn] = 2. On 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 R2 (1 lecture)

Dihedral and cyclic groups as symmetry and direct symmetry groups of a regular n-gon. Basic properties of Dn. Finite subgroups of O2 are dihedral or cyclic.

3. Wallpaper patterns I: the isometry group of R2. (2 lectures)

Group of isometries and groups of translations of Rn, In and Tn. Tn @ (Rn,+). Examples in R2: reflections, rotations, glides. An isometry of Rn fixing O is linear. I2 consists of translations, rotations, reflections and glides. An isometry of Rn is determined by its values on xi, i = 0,...,n if (x1-x0, ... , xn-x0) are linearly independent.

4. Semidirect products. (2 lectures).

Semidirect product, definition and characterisation. Examples to include Dn = Cn \rtimes S and In = Tn\rtimesOn. Dn 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 R3.

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; Dn as a subgroup of Sn, for n 3.

Reflections in a hyperplane. Every rotation in SO3 has an axis.

The tetrahedron.(1/2 lecture)

Faithful action on vertices. Symm(Tet) @ S4. If G On then [G:GSOn] 2. Dir(Tet) @ A4.

The cube. (1/2 lecture)

Faithful action of Dir(Cube) on long diagonals. Description of 24 rotations in Dir(Cube). Dir(Cube) @ S4. G O3 and -1 G implies that G @ GSO3×{1}. Symm(Cube) @ S4×{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) @ A5. Symm(Dodec ) @ A5×{1}. A5 only proper, nontrivial normal subgroup of S5.

Finite rotation groups in R3 (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 SO3 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 S4). Any group of order pq (with p q) is a semidirect product. Proof of Sylow theorems. |H||K| = |HK||HK|, 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(Cr) as the group of units of \mathbbZr 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, Q12. The classification of groups of order 15.

10. Finite simple groups. (2 lectures) Simple groups. The classification. SL2(Z/p). Subgroups of order p and p-1. The centre of SL2(Z/p). The group PSL2(Z/p). The simplicity of PSL2(Z/p) for p 5; proof for p = 7.

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On 28 Sep 1999, 16:03.