Examination topics

1. Group axioms; homomorphisms and isomorphisms; conjugacy classes.

2. Subgroups; system of generators; cyclic groups.

3. Cosets; Lagrange's theorem; normal subgroups and their factor groups.

4. The homomorphism theorem; free groups and the Nielsen-Schreier theorem; group presentations.

5. Direct product of groups; the Frobenius-Stickelberger theorem.

6. Derived series and soluble groups; composition series and the Jordan-Hölder theorem.

7. Group representations; invariant subspaces and reducibility; Schur's lemma.

8. Direct sum of representations; theorems of Maschke and  Peter-Weyl; irreducible decomposition.

9. Tensor product of representations; fusion rules.

10. Lie groups and  Lie algebras; local isomorphism; the universal covering group.

11. The rotation group and its Lie algebra; relation to SU(2).