https://teddyrookbook.com/math/abstract-algebra/Recent content in Abstract Algebra on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/abstract-algebra/binary-operations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/binary-operations/A binary operation combines two elements of a set into one element of the set; a Cayley table lists every product a ∗ b (row a, column b), so you can read off closure, an identity and commutativity. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/group-axioms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/group-axioms/A group is a set with an associative operation, an identity and an inverse for every element; decide which axiom fails for examples like ℤ under subtraction or ℤₙ under multiplication. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/units-mod-n/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/units-mod-n/ℤₙ is {0, 1, …, n − 1} under addition mod n; U(n) is the set of residues coprime to n under multiplication mod n, a group of order φ(n). Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/group-properties/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/group-properties/From the axioms alone: the identity and inverses are unique, ab = ac gives b = c, and (ab)⁻¹ = b⁻¹a⁻¹. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/element-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/element-order/The order of a is the least n ≥ 1 with aⁿ = e (n·a = 0 when the operation is addition), and aᵏ = e exactly when the order of a divides k. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/subgroups/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/subgroups/H is a subgroup when it is a group under the same operation; it is enough that H is nonempty and ab⁻¹ ∈ H for all a, b ∈ H (for a finite H, closure alone is enough). Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/cyclic-groups/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/cyclic-groups/⟨a⟩ = {aᵏ : k ∈ ℤ} is the smallest subgroup containing a; G is cyclic when one element generates it, and k generates ℤₙ exactly when gcd(k, n) = 1. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/order-of-powers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/order-of-powers/If a has order n, then aᵏ has order n / gcd(n, k); in ℤₙ the element k has order n / gcd(n, k). Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/subgroups-of-cyclic/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/subgroups-of-cyclic/Every subgroup of a cyclic group is cyclic; ℤₙ has exactly one subgroup of each order d dividing n, namely ⟨n/d⟩, and it has φ(d) elements of order d. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/subgroup-lattice/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/subgroup-lattice/Draw the subgroups of a group ordered by inclusion; for ℤₙ the picture is the divisor diagram of n turned upside down, and you read off intersections and joins. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/permutations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/permutations/A permutation is a bijection from {1, …, n} to itself; Sₙ is the group of all n! of them under composition, and στ means do τ first, then σ. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/cycle-notation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/cycle-notation/Follow 1 → σ(1) → σ(σ(1)) → … until you come back, then start a new cycle with the smallest unused number; disjoint cycles commute, and overlapping cycles must be multiplied out. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/permutation-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/permutation-order/Write σ as disjoint cycles: its order is the lcm of the cycle lengths, and its cycle type tells how many permutations share its shape. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/sign-alternating/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/sign-alternating/Every permutation is a product of transpositions, always an even number or always an odd number of them; a k-cycle is even when k is odd, and the even permutations form Aₙ, of order n!/2. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/dihedral-groups/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/dihedral-groups/Dₙ is the group of the 2n symmetries of a regular n-gon: rotations rᵏ and reflections rᵏs, with rⁿ = s² = e and sr = r⁻¹s. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/cosets/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/cosets/The left coset aH = {ah : h ∈ H}; the left cosets of H all have |H| elements and split G into disjoint pieces, and aH = bH exactly when a⁻¹b ∈ H. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/lagrange/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/lagrange/If H is a subgroup of a finite group G, then |H| divides |G|, and the index [G : H] = |G|/|H| counts the cosets. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/lagrange-consequences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/lagrange-consequences/The order of every element divides |G|, so a^|G| = e; a group of prime order is cyclic; and Fermat's and Euler's theorems follow by taking G = U(n). Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/normal-subgroups/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/normal-subgroups/N is normal when gNg⁻¹ = N for every g, which is the same as gN = Ng; subgroups of abelian groups and subgroups of index 2 are always normal. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/quotient-groups/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/quotient-groups/When N is normal, the cosets of N form a group G/N with (aN)(bN) = abN, of order |G|/|N|. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/homomorphisms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/homomorphisms/φ : G → H is a homomorphism when φ(ab) = φ(a)φ(b); then φ(e) = e, φ(a⁻¹) = φ(a)⁻¹, and the order of φ(a) divides the order of a. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/kernels-images/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/kernels-images/The kernel {g : φ(g) = e} is a normal subgroup, the image is a subgroup, φ is one-to-one exactly when the kernel is {e}, and |G| = |ker φ| · |im φ|. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/isomorphisms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/isomorphisms/An isomorphism is a bijective homomorphism; to show two groups are not isomorphic, find a property one has and the other lacks: order, being abelian or cyclic, or how many elements have each order. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/cayley-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/cayley-theorem/Every group is isomorphic to a group of permutations: g acts on G by x ↦ gx, so a group of order n sits inside Sₙ. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/first-isomorphism/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/first-isomorphism/If φ : G → H is a homomorphism, then G/ker φ ≅ im φ; use it to name quotients such as ℤ/nℤ ≅ ℤₙ and Sₙ/Aₙ ≅ ℤ₂. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/direct-products/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/direct-products/In G × H the operation works coordinate by coordinate; the order of (g, h) is lcm(|g|, |h|), and ℤₘ × ℤₙ ≅ ℤₘₙ exactly when gcd(m, n) = 1. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/finite-abelian/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/finite-abelian/Every finite abelian group is a product of cyclic groups of prime-power order; for order pᵃ the choices match the partitions of a, so you can list them all. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/invariant-factors/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/invariant-factors/Regroup a product of cyclic groups so each order divides the next (ℤ₂ × ℤ₄ × ℤ₃ ≅ ℤ₂ × ℤ₁₂), and count elements of each order to tell abelian groups apart. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/group-actions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/group-actions/A group acts on a set X when each g moves points, x ↦ g·x, with e·x = x and g·(h·x) = (gh)·x; the orbit of x is everywhere it can go, and its stabilizer is the subgroup that fixes it. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/orbit-stabilizer/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/orbit-stabilizer/For a finite group, |orbit of x| · |stabilizer of x| = |G|, because the points of the orbit match the cosets of the stabilizer. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/class-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/class-equation/Conjugacy classes are the orbits of G acting on itself by x ↦ gxg⁻¹; in Sₙ they are the cycle types, and |G| = |Z(G)| + the sum of [G : C(x)] over the classes with more than one element. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/burnside/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/burnside/The number of orbits is the average number of points fixed by a group element; count necklaces and colorings up to symmetry by adding up the fixed colorings over the group. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/sylow/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/sylow/If |G| = pᵃm with p not dividing m, then G has subgroups of order pᵃ, they are all conjugate, and their number nₚ divides m and is ≡ 1 (mod p). Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/rings/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/rings/A ring is an abelian group under addition with an associative multiplication that distributes over it; ℤ, ℤₙ, matrices and polynomials are rings, and the units are the elements with multiplicative inverses. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/integral-domains/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/integral-domains/A zero divisor is an a ≠ 0 with ab = 0 for some b ≠ 0; an integral domain has none, every field is one, and ℤₙ is a field exactly when n is prime. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/ideals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/ideals/An ideal is an additive subgroup I with ra and ar in I for every r in R and a in I; nℤ and ⟨d⟩ in ℤₙ are ideals, and every ideal of ℤ has the form nℤ. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/ring-homomorphisms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/ring-homomorphisms/φ respects both operations, φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b); its kernel is an ideal, and reduction ℤ → ℤₙ and evaluation f ↦ f(c) are the key examples. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/quotient-rings/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/quotient-rings/For an ideal I, the cosets a + I form a ring R/I; compute in ℝ[x]/(x² + 1), where x² can be replaced by −1, and in ℤₚ[x]/(f) by taking remainders. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/prime-maximal-ideals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/prime-maximal-ideals/P is prime when ab ∈ P forces a ∈ P or b ∈ P, and M is maximal when no ideal lies strictly between M and R; R/P is an integral domain and R/M is a field. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/polynomial-rings/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/polynomial-rings/R[x] holds the polynomials with coefficients in R; over an integral domain deg(fg) = deg f + deg g, but in ℤₙ[x] with n composite degrees can drop. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/polynomial-division/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/polynomial-division/Over a field, f = qg + r with deg r < deg g; c is a root of f exactly when x − c divides f, so a nonzero polynomial of degree n has at most n roots. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/polynomial-gcd/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/polynomial-gcd/Repeated division finds gcd(f, g) in F[x], and running it backward writes the gcd as af + bg, which is how inverses are found in F[x]/(p(x)). Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/irreducible-low-degree/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/irreducible-low-degree/A polynomial of degree 2 or 3 over a field is irreducible exactly when it has no root in the field, so over ℤₚ you can test every element. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/irreducibility-tests/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/irreducibility-tests/Use the rational root test, Gauss's lemma, Eisenstein's criterion (a prime dividing every coefficient but the leading one, with p² not dividing the constant) and reduction mod p. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/field-extensions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/field-extensions/If p(x) is irreducible over F, then F[x]/(p(x)) is a field containing a root of p; ℚ(√2) = {a + b√2 : a, b ∈ ℚ} is one, with inverses found by the conjugate. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/minimal-polynomials/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/minimal-polynomials/The minimal polynomial of α over F is the monic irreducible polynomial with α as a root, and [F(α) : F] equals its degree. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/tower-law/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/tower-law/For fields F ⊆ K ⊆ L, [L : F] = [L : K][K : F]; so [ℚ(√2, √3) : ℚ] = 4, and the cube root of 2 lies in no extension of ℚ whose degree is a power of 2. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/finite-fields/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/finite-fields/GF(pⁿ) = ℤₚ[x]/(f) for an irreducible f of degree n: add and multiply polynomials mod p and mod f, and find inverses with the Euclidean algorithm. Abstract Algebra · Math Ladderhttps://teddyrookbook.com/math/abstract-algebra/finite-field-structure/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/abstract-algebra/finite-field-structure/The nonzero elements of GF(pⁿ) form a cyclic group of order pⁿ − 1, generated by a primitive element, and GF(pⁿ) contains GF(pᵐ) exactly when m divides n. Abstract Algebra · Math Ladder