Section outline

    1. Introduction and basics
      Definition of group, subgroup, order, generators, and basic properties.
    2. Examples
      Cyclic gropus, the quaternion group of order 8, symmetric groups, alternating groups, symmetry group of geometric objects, matrix groups, group of units modulo n.
    3. Cosets and Conjugacy
      Cosets, Lagrange's Theorem, conjugacy, normal subgroups, quotient groups, products of subgroups, the commutator subgroup.
    4. Homomorphisms
      Definitions, image and kernel, the Isomorphism Theorems, the Correspondence Theorem, automorphisms, inner and outer automorphism groups.
    5. Actions
      Definitions, orbits, stabilisers, the Orbit–Stabiliser Theorem, centralisers and normalisers, the Orbit-Counting Lemma and applications.
    6. Simple groups and composition series
      Definitions, simple abelian groups, simplicity of alternating groups, composition series, statement of Jordan-Hölder.
    7. p-groups
      Sylow p-subgroups, the Sylow theorems, finite p-groups.