| The most important thing is to work the problems on the Exercise List. Also, be sure you have internalized the following definitions, results, and ideas. It is more important to grasp the meanings and consequences of the results listed below than to be able to prove them. (But you should be able to supply at least an outline of most proofs.) You should be able to give a paraphrased definition of any term listed below. |
1.1: Basic terms: Group, abelian, order of an element. |
1.2: Basic terms: Dihedral groups, generators, relations. |
1.3: Basic terms: Symmetric groups, cycles, cycle decomposition; basic properties of cycles. |
1.4: Basic terms: Matrix groups over a field; general linear group. |
1.5: Basic terms: Quaternions. |
1.6: Basic terms: Homomorphisms, isomorphisms and their basic properties. |
1.7: Basic terms: Group actions, kernels of actions, permutation representations, faithful actions. |
2.1: Basic terms: Subgroups and subgroup criteria. |
2.2: Basic terms: Centralizers, normalizers, stabilizers and kernels. |
2.3: Basic terms and results: Cyclic groups, Proposition 2, Theorem 4, Proposition 5, Proposition 6, Theorem 7. |
2.4: Basic terms and results: Proposition 8, Proposition 9. |
2.5: Basic terms: Subgroup lattices, Klein 4-group. |
3.1: Basic terms and results: Proposition 1, Proposition 2, Theorem 3, Proposition 4, Proposition 5, Theorem 6, Proposition 7, natrural projection homomorphism. |
3.2: Basic terms and results: Theorem 8, Corollary 9, Corollary 10, Theorem 11. |
3.3: Basic terms and results: Theorem 16, Corollary 17. |
3.4: (not represented on test) |
3.5: Basic terms and results: Transpositions, alternating groups, sign of a permutation, even and odd permutations. |
4.1: Basic terms and results: Orbits, permutation representation, Proposition 2 (that is, orbits form a partition). |
4.2: Basic result: Theorem 3. |
4.3: Basic terms and results: Conjugation, conjugacy class, cycle type, Proposition 6, Theorem 7, Proposition 10, Proposition 11, Theorem 12. |
4.4: Basic terms and results: Automorphism, inner automorphism, characteristic subgroups, Proposition 13, Corollary 14, Corollary 15, Proposition 16. |
4.5: Basic terms and results: Sylow p-subgroup, Theorem 18, Corollary 20 |
4.6: Basic result: Theorem 24 (An is simple for n > 4.) |
The final exam will cover the following material: |
5.1: Basic terms and results: Propositions 1 and 2 |
5.2: Basic terms and results: Theorem 3, Corollary 4, Theorem 5, Proposition 6 |
5.4: Basic terms and results: Theorem 9 |
5.5: Basic terms and results: Theorem 10, Proposition 11, Theorem 12 |
7.1: Basic terms and results: Rings, zero divisors, units, rings with identities, division rings, fields, Propositions 1, 2; Corollary 3, |
7.2: Basic terms and results: Basic examples of rings (excluging group rings), Proposition 4. |
7.3: Basic terms and results: Propositions 5, 6; Theorem 7 |
7.4: Basic terms and results: Ideals, prime ideals, maximal ideals, principal ideals, Proposition 9; Corollary 10; Propositions 11, 12, 13; Corollary 14 |
7.6: Basic terms and results: Theorem 17, Corollary 18 |
8.1: Basic terms and results: Euclidean domains, Propositions 1, 2, 3; Theorem 4 |
8.2: Basic terms and results: Propositions 6, 7; Corollary 8; |
8.3: (Not on exam.) |