Abstract Algebra

Units

1. 1

   Groups

   3 lessons

   1. Binary operations and groupsState the four group axioms (closure, associativity, identity, inverses) and check or refute them for ℤ under addition, ℤ₆, the nonzero rationals and 2 × 2 invertible matrices.
   2. Symmetry groupsList the eight symmetries of a square as r (a quarter turn) and s (a reflection), build their Cayley table, and see the rule rs = sr⁻¹ that governs the dihedral group Dₙ.
   3. Basic properties and orderProve that the identity and inverses are unique, that cancellation holds, that (ab)⁻¹ = b⁻¹a⁻¹, and find the order of an element in ℤ₁₂ and in D₄.
2. 2

   Subgroups and cyclic groups

   2 lessons

   1. SubgroupsUse the one-step subgroup test, show that the subgroups of ℤ are the sets nℤ, find the centre of D₄, and read the subgroup lattice of ℤ₁₂.
   2. Cyclic groupsDefine generators and cyclic groups, show every cyclic group is like ℤ or ℤₙ, prove that subgroups of cyclic groups are cyclic, and find the generators of ℤₙ as the k with gcd(k, n) = 1.
3. 3

   Permutation groups

   2 lessons

   1. Permutations and cycle notationPermutations of {1, ..., n} as arrow diagrams and in cycle notation, composing right to left, disjoint cycles and the order as the lcm of cycle lengths, in S₃ and S₄.
   2. Even and odd permutationsEvery permutation is a product of transpositions with a fixed parity; the alternating group Aₙ has half the elements of Sₙ.
4. 4

   Cosets and Lagrange's theorem

   2 lessons

   1. Cosets and Lagrange's theoremLeft cosets of a subgroup partition the group into equal pieces, so the order of a subgroup divides the order of the group; drawn for the rotation subgroup of D₄.
   2. Consequences of Lagrange's theoremThe order of an element divides |G|, groups of prime order are cyclic, and Fermat's little theorem and Euler's theorem from the group of units mod n.
5. 5

   Homomorphisms and quotient groups

   3 lessons

   1. Homomorphisms and isomorphismsMaps that preserve the operation, kernel and image, isomorphic groups (Z₄ and the rotations of a square; Z₂ × Z₂ is not Z₄).
   2. Normal subgroups and quotient groupsNormal subgroups as those whose left and right cosets agree, the quotient group G/N of cosets, worked on Z/4Z and D₄ modulo its centre.
   3. The first isomorphism theoremG/ker φ ≅ φ(G), applied to the determinant on invertible matrices, to reduction Z → Zₙ and to the sign of a permutation.
6. 6

   Direct products and group actions

   2 lessons

   1. Direct productsG × H with componentwise operation, Zₘ × Zₙ ≅ Zₘₙ exactly when gcd(m, n) = 1, and the statement of the classification of finite abelian groups.
   2. Group actionsA group acting on a set, orbits and stabilizers, the orbit-stabilizer theorem |G| = |orbit| × |stabilizer|, and counting distinct colourings of a square's corners with Burnside's lemma.
7. 7

   Rings

   4 lessons

   1. RingsThe ring axioms on Z, Zₙ, polynomials and 2 × 2 matrices, commutative rings, units and rings with identity.
   2. Integral domains and fieldsZero divisors (2 · 3 = 0 in Z₆), integral domains, fields, every finite integral domain is a field, and Zₙ is a field exactly when n is prime, shown by multiplication tables.
   3. Ideals and quotient ringsIdeals as the kernels of ring maps, principal ideals, the quotient ring R/I, prime and maximal ideals, and Z/nZ.
   4. Ring homomorphismsRing maps, kernel as an ideal, the first isomorphism theorem for rings, and R[x]/(x² + 1) ≅ C by evaluation at i.
8. 8

   Polynomial rings and factorization

   3 lessons

   1. Polynomial rings and divisionF[x] over a field, the division algorithm worked in Q[x] and Z₅[x], the remainder and factor theorems, and gcds of polynomials.
   2. Irreducible polynomialsIrreducibility over Q, R, C and Zₚ, the rational root test, Eisenstein's criterion and reduction mod p, and degree 2 and 3 tests by roots.
   3. Unique factorizationEuclidean domains, principal ideal domains and unique factorization domains, with the Gaussian integers Z[i] and the failure in Z[√−5] where 6 = 2 · 3 = (1 + √−5)(1 − √−5).
9. 9

   Fields

   2 lessons

   1. Field extensionsQ(√2) as a vector space over Q, the degree of an extension, the minimal polynomial, F[x]/(p(x)) as a field when p is irreducible, and the tower law.
   2. Finite fieldsA finite field has pⁿ elements; constructs the field of 4 elements as Z₂[x]/(x² + x + 1) with its addition and multiplication tables, and its multiplicative group is cyclic.