Group Theory

MTH1201 Foundation of Mathematics

3 hours No requisites

1 The Concept of Groups: Definitions, Examples, and Basic Properties: Definition of groups, the order of the group, finite and infinite groups examples of groups: Dihedral groups, Quaternion groups, group of matrices (with addition), General linear groups and special linear groups, Caley tables of groups and Abelian groups. / Present basic concepts and properties of groups as: The identity is unique, the inverse is unique, the cancellation laws hold, inverse of the product of two elements

2 Subgroups: / Definition of subgroup of a group, examples of subgroups, intersection and union of subgroups. The center of the group and the centeralizer of the element in the group

3 Some Concrete Examples of Groups: / Definition of permutation, composition of permutation, cycles, disjoint permutations, decomposition of permutation as cycles, transpositions, identity permutation, Symmetric groups and permutation groups, alternating groups as a permutation group

4 Cyclic and Abelian groups: / Definition of the order of an element in the group, definition and examples of cyclic groups. Relationship between cyclic groups and Abelian groups. Examples of an abelian group which is not cyclic

5 Normal Subgroups and Quotient (Factor) Groups : / Definition of the cosets of subgroups, definition and examples of normal subgroups, criteria for subgroup to be normal. Lagrange Theorem, the index of a subgroup, partition of a group as cosets. Definition and examples of quotient groups. Properties of the quotient groups

6 Group Homomorphism: / Definition and examples of group homomorphism, isomorphism and automorphism. First, second, third isomorphism theorems of groups. Conjugacy classes and class equation

7 Direct and Semidirect Product of Groups: / Definition and examples of direct and semidirect product of groups

8 Others: Revision and activities, Exam, quizzes

1.1 Mapped to: K1

Develop the ability to articulate definitions, invoke relevant facts, provide illustrative examples and insightful counterexamples, and effectively apply the properties of groups, subgroups, cyclic groups, normal subgroups, quotient groups, permutation groups, and group actions.

Teaching Strategy Lectures