Lancaster University

• Conjugacy; conjugacy classes; centralizers; conjugacy and normal subgroups; conjugacy for permutations.
• External and internal direct products;classification of finite abelian groups.
• Group actions; orbits and stabilizers; the orbit-stabilizer theorem; classification and symmetry groups of Platonic solids.
• Series of groups; Jordan-Holder theorem; simplicity of $A_n$
• Sylow's theorems and applications; elementary results on p-groups.

• The classification of finite abelian groups.
• The orbit-stabilizer theorem.
• The Jordan-Holder theorem.
• The classification and symmetry groups of the Platonic solids.
• Sylow's theorems.

We shall first consider a way of comparing the elements of a group and show how a group may be built up from smaller components using 'direct products'. Next we shall treat situations in whch a group 'acts' on a set by permuting its elements; after identifying the five Platonic solids, we shall use group actions to determine their symmetry groups. Finally we shall prove some interesting and important results, known as the 'Sylow theorems', relating to subgroups of certain orders.

• The classification of finite abelian groups
• The orbit-stabilizer theorem
• the Jordan-Holder theorem
• The classification and symmetry groups of the Platonic solids
• Sylow's theorems.

• understand the concept of conjugacy in groups and its applications, especially for groups of permutations;
• be able to state and apply the classification theorem for finite abelian groups;
• understand composition sseries and be able to compute them for examples of finite groups
• understand the proof the simplicity of $A_n$
• understand group actions;
• know the five Platonic solids and understand how their symmetry groups are determined;
• be able to state, understand the proofs of, and apply Sylow's theorems.

• understand the concept of conjugancy in groups and its applications, especially for groups of permutations;
• be able to state and apply the classification theorem for finite abelian groups
• understand composition series and be able to compute them for examples of finite group
• understand the proof the simplicity of $A_n$
• understand group actions
• know the five Platonic solids and understand how their symmetry groups are determined
• be able to state, understand the proofs of, and apply Sylow's theorems

• B Fraleigh, A First Course in Abstract Algebra (seventh edition), Addison-Wesley, 2003.
• J A Gallian, Contemporary Abstract Algebra (fourth edition), Houghton-Mifflin, 1998.
• F M Goodman, Algebra; Abstract and Concrete, Prentice-Hall, 1998.
• D A R Wallace, Groups, Rings and fields (second printing), Springer-Verlag, 2001.

1. as a core/compulsory module
2. as an optional module
3. Msci/BSc/BA degrees in Mathematics and Statistics single and combined, CSMA001 Discrete Mathematics