Groups

Module aims

Prove elementary structural theorems from axioms and keep examples (cyclic, symmetric, matrix) as illustrations, not as substitutes for proofs.

Course outline

1. 01 Group axioms and examples 15h

   Learning result Check axioms on a stated set-with-operation, including a non-example.

   Formative check Counterexample

2. 02 Subgroups and cyclic groups 15h

   Learning result Prove the subgroup criterion and classify cyclic groups of a given order up to the course’s stated correspondence.

   Formative check Written derivation

3. 03 Homomorphisms and Lagrange 20h

   Learning result Prove that the kernel is a subgroup and apply Lagrange to bound element orders.

   Formative check Written derivation

4. 04 Cosets and quotient groups 20h

   Learning result Construct a quotient by a normal subgroup and give a non-normal subgroup where the set of cosets is not a group under the naive operation.

   Formative check Counterexample

5. 05 Group actions (introduction) 15h

   Learning result State an action, an orbit, and a stabiliser, and apply orbit-stabiliser in a small example.

   Formative check Four-option check

6. 06 Synthesis 15h

   Learning result Write one structural argument that uses a homomorphism and Lagrange together.

   Formative check Written derivation

Learning results

Verify the group axioms, compute a subgroup or kernel, and apply Lagrange with its hypotheses.

Subject skills

Axioms, cyclic groups, homomorphisms, Lagrange, quotient groups at first-course grain.

Transferable skills

Test a claimed identity on a non-abelian example; write a homomorphism proof in kernel-image language.