Groups

模块目标

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

课程大纲

1. 01 Group axioms and examples 15h

   学习结果 Check axioms on a stated set-with-operation, including a non-example.

   形成性检查 反例

2. 02 Subgroups and cyclic groups 15h

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

   形成性检查 书面推导

3. 03 Homomorphisms and Lagrange 20h

   学习结果 Prove that the kernel is a subgroup and apply Lagrange to bound element orders.

   形成性检查 书面推导

4. 04 Cosets and quotient groups 20h

   学习结果 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.

   形成性检查 反例

5. 05 Group actions (introduction) 15h

   学习结果 State an action, an orbit, and a stabiliser, and apply orbit-stabiliser in a small example.

   形成性检查 四选一检查

6. 06 Synthesis 15h

   学习结果 Write one structural argument that uses a homomorphism and Lagrange together.

   形成性检查 书面推导

学习结果

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

学科技能

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

可迁移技能

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