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).