Abstract Algebra: Groups, Rings, Fields, and Galois Theory
Abstract algebra studies algebraic structures — groups, rings, fields, and modules — by isolating their essential properties and reasoning about them axiomatically. It is the language of modern mathematics and the backbone of cryptography, coding theory, and physics.
1. Groups: Axioms, Examples, and Subgroups
A group is the simplest algebraic structure that captures the notion of symmetry. Formally, a group is a set G together with a binary operation (often written as multiplication or addition) satisfying four axioms.
The Group Axioms
- Closure:For all a, b in G, the product ab is in G.
- Associativity:For all a, b, c in G, we have (ab)c = a(bc).
- Identity:There exists e in G such that ea = ae = a for all a in G.
- Inverses:For each a in G, there exists an element a-inverse in G with a times a-inverse equal to e.
When ab = ba for all a, b in G, the group is called abelian (or commutative). Non-abelian groups are the rule rather than the exception in higher mathematics.
Fundamental Examples
Integers under Addition (Z, +)
The integers form an abelian group under addition. The identity is 0, the inverse of n is -n, and closure and associativity are immediate. This is an infinite group of infinite order.
Integers mod n (Z/nZ)
The integers modulo n form a cyclic group of order n under addition mod n. When n is prime, the nonzero elements also form a group under multiplication, making Z/pZ a field.
Symmetric Group S sub n
The symmetric group on n elements consists of all permutations of a set of n elements under composition. It has order n factorial. S sub 3 is the smallest non-abelian group, with order 6.
Dihedral Group D sub n
The dihedral group D sub n is the symmetry group of a regular n-gon, consisting of n rotations and n reflections. It has order 2n and is non-abelian for n greater than or equal to 3.
General Linear Group GL(n, F)
The general linear group GL(n, F) consists of all invertible n-by-n matrices with entries in a field F, under matrix multiplication. It is non-abelian for n at least 2. The special linear group SL(n, F) is the subgroup of matrices with determinant equal to 1.
Subgroups
A subset H of a group G is a subgroup if H is itself a group under the same operation. The subgroup test gives a convenient criterion: H is a nonempty subset of G that is closed under the group operation and closed under taking inverses. Equivalently, H is nonempty and for all a, b in H, the element a times b-inverse is in H.
Important Subgroup Examples
- The even integers form a subgroup of (Z, +).
- The alternating group A sub n (even permutations) is a subgroup of S sub n of index 2.
- The center Z(G) consists of all elements that commute with every element of G; it is always a normal abelian subgroup.
- The cyclic subgroup generated by an element a is the set of all powers of a: e, a, a squared, a cubed, and so on.
Cosets and Lagrange's Theorem
If H is a subgroup of G and a is an element of G, the left coset aH is the set of all products ah where h ranges over H. The collection of left cosets of H partitions G into equal-sized pieces, each of cardinality equal to the order of H.
Lagrange's Theorem
If G is a finite group and H is a subgroup of G, then the order of H divides the order of G. The index [G:H] = |G| divided by |H| equals the number of distinct left cosets of H in G. As a corollary, the order of any element g (the smallest positive integer k with g to the k equal to e) divides |G|, and therefore g to the |G| equals e for every g in G.
2. Group Homomorphisms and the Isomorphism Theorems
A group homomorphism is a structure-preserving map between groups. Homomorphisms, their kernels, and the isomorphism theorems are the core tools for understanding relationships between groups.
Homomorphisms, Kernels, and Images
A function f from G to H is a group homomorphism if f(ab) = f(a)f(b) for all a, b in G. The kernel of f is the set of all elements of G that map to the identity of H; it is always a normal subgroup of G. The image of f is the set of all values f(g) as g ranges over G; it is always a subgroup of H.
Key Properties
- f sends the identity of G to the identity of H.
- f sends inverses to inverses: f(a-inverse) = f(a)-inverse.
- f is injective if and only if the kernel of f is trivial (contains only the identity).
- An isomorphism is a bijective homomorphism; its inverse is also a homomorphism.
Normal Subgroups and Quotient Groups
A subgroup N of G is normal if gNg-inverse equals N for every g in G, equivalently if every left coset equals the corresponding right coset. Normal subgroups are precisely the kernels of homomorphisms. When N is normal in G, the set of cosets G/N forms a group called the quotient group, with multiplication defined by (aN)(bN) = (ab)N.
The Isomorphism Theorems
First Isomorphism Theorem
If f: G to H is a group homomorphism with kernel K, then K is normal in G and G/K is isomorphic to the image of f. The isomorphism sends the coset gK to f(g). Every surjective homomorphism from G to H realizes H as a quotient of G.
Second Isomorphism Theorem
If H is a subgroup of G and N is a normal subgroup of G, then HN is a subgroup of G, H intersect N is a normal subgroup of H, and H/(H intersect N) is isomorphic to HN/N.
Third Isomorphism Theorem
If K is normal in G and K is contained in N which is normal in G, then N/K is normal in G/K and (G/K)/(N/K) is isomorphic to G/N. This allows cancellation of normal subgroups in quotients.
3. Cyclic Groups and Finitely Generated Abelian Groups
Cyclic groups are the simplest groups and serve as the building blocks for all finitely generated abelian groups. Their complete classification is one of the first major theorems in the subject.
Cyclic Groups and Generators
A group G is cyclic if there exists an element g in G such that every element of G is a power of g. Such an element g is called a generator of G. The cyclic group of order n is isomorphic to Z/nZ, and the infinite cyclic group is isomorphic to (Z, +).
Key Facts About Cyclic Groups
- Every subgroup of a cyclic group is cyclic.
- The cyclic group Z/nZ has exactly one subgroup of each order dividing n, and these are all of its subgroups.
- The number of generators of Z/nZ equals Euler's totient function phi(n), the count of integers from 1 to n that are coprime to n.
- Every group of prime order is cyclic (and hence abelian).
Classification of Finitely Generated Abelian Groups
The fundamental theorem of finitely generated abelian groups states that every finitely generated abelian group is isomorphic to a direct product of cyclic groups. More precisely, there are two equivalent normal forms:
Invariant Factor Decomposition
G is isomorphic to Z/d1 times Z/d2 times ... times Z/dk times Z to the r, where d1 divides d2, d2 divides d3, and so on. The integers d1, d2, ..., dk are the invariant factors of G and r is the free rank (or Betti number).
Primary Decomposition
G is isomorphic to a direct product of cyclic groups of prime power order, one product for each prime p dividing the order of the torsion part of G. The primary components are uniquely determined up to isomorphism.
Example: Groups of Order 12
The abelian groups of order 12 are: Z/12Z and Z/2Z times Z/6Z (equivalently Z/2Z times Z/2Z times Z/3Z). These are non-isomorphic because Z/12Z is cyclic (has an element of order 12) while Z/2Z times Z/6Z has no element of order 12.
4. Symmetric Groups: Cycles, Signs, and Conjugacy
The symmetric group S sub n on n elements plays a central role in algebra. By Cayley's theorem, every finite group embeds in some symmetric group, making S sub n the universal finite group.
Cycle Notation
Every permutation in S sub n can be written uniquely as a product of disjoint cycles. A k-cycle (a1 a2 ... ak) sends a1 to a2, a2 to a3, ..., and ak back to a1, while fixing all other elements. Disjoint cycles commute with each other.
Cycle Type and Conjugacy
The cycle type of a permutation is the partition of n given by its cycle lengths. Two permutations in S sub n are conjugate if and only if they have the same cycle type. Therefore, conjugacy classes in S sub n correspond to partitions of n. For example, in S sub 4, the conjugacy classes correspond to the partitions: 1+1+1+1, 2+1+1, 2+2, 3+1, and 4.
The Sign Homomorphism and the Alternating Group
Every permutation can be written as a product of transpositions (2-cycles). Although this decomposition is not unique, the parity (even or odd number of transpositions) is well-defined. The sign homomorphism sgn: S sub n to the multiplicative group with elements 1 and -1 sends even permutations to 1 and odd permutations to -1.
The alternating group A sub n is the kernel of the sign homomorphism: it consists of all even permutations and has order n factorial divided by 2. A sub n is a normal subgroup of S sub n of index 2. For n at least 5, A sub n is a simple group, a fact that is central to the proof of the Abel-Ruffini theorem.
Signs of Cycle Types
- A k-cycle has sign (-1) to the power (k-1).
- Odd-length cycles (3-cycles, 5-cycles, ...) are even permutations.
- Even-length cycles (transpositions, 4-cycles, ...) are odd permutations.
- The sign of a product of disjoint cycles is the product of the signs of the individual cycles.
5. The Sylow Theorems and Applications
The Sylow theorems are the most powerful tools in finite group theory. They generalize the converse of Lagrange's theorem for prime power divisors and give detailed information about the subgroup structure of finite groups.
p-Groups and Sylow p-Subgroups
A p-group is a group in which every element has order a power of the prime p. A finite group is a p-group if and only if its order is a power of p. If the order of G is p to the k times m with p not dividing m, then a Sylow p-subgroup is a subgroup of order p to the k (the highest power of p dividing |G|).
The Three Sylow Theorems
Sylow I: Existence
If p to the k divides the order of G, then G has a subgroup of order p to the k. In particular, Sylow p-subgroups of order p to the k always exist.
Sylow II: Conjugacy
All Sylow p-subgroups of G are conjugate to each other. If P and Q are both Sylow p-subgroups of G, then there exists g in G such that Q = gPg-inverse. In particular, if there is only one Sylow p-subgroup, it is normal in G.
Sylow III: Number of Sylow Subgroups
The number of Sylow p-subgroups, written n sub p, satisfies two congruence conditions: n sub p divides m (the part of |G| not divisible by p), and n sub p is congruent to 1 modulo p. These constraints often force n sub p to equal 1, yielding a normal Sylow subgroup.
Applications: Classifying Groups of Small Order
Example: Groups of Order 15
Let |G| = 15 = 3 times 5. By Sylow III, n sub 5 divides 3 and n sub 5 is congruent to 1 mod 5, so n sub 5 equals 1. Similarly n sub 3 divides 5 and n sub 3 is congruent to 1 mod 3, so n sub 3 equals 1. Both Sylow subgroups are normal, and since their orders are coprime, G is isomorphic to their direct product Z/3Z times Z/5Z, which is isomorphic to Z/15Z. Therefore every group of order 15 is cyclic.
6. Rings, Ideals, and Ring Homomorphisms
A ring is an algebraic structure that generalizes the integers. It has two operations — addition and multiplication — with addition forming an abelian group and multiplication distributing over addition.
Ring Axioms
A ring (R, +, ·) satisfies:
- (R, +) is an abelian group with identity 0.
- Multiplication is associative: a(bc) = (ab)c.
- Distributivity holds: a(b+c) = ab + ac and (a+b)c = ac + bc.
A ring with a multiplicative identity element 1 is called unital. A commutative unital ring with no zero divisors is an integral domain. A commutative ring in which every nonzero element is a unit is a field.
Ideals
An ideal I of a ring R plays the same role for rings that normal subgroups play for groups. A left ideal I satisfies: I is a subgroup of (R, +), and for all r in R and a in I, the product ra is in I. A two-sided ideal satisfies both ra in I and ar in I for all r in R. Ideals are exactly the kernels of ring homomorphisms.
Principal Ideals
The principal ideal generated by an element a, written (a) or aR, is the set of all multiples of a. In Z, every ideal is principal: the ideals are exactly the sets nZ for non-negative integers n. A ring in which every ideal is principal is called a principal ideal domain (PID).
Prime and Maximal Ideals
An ideal P is prime if whenever ab is in P, then a is in P or b is in P. P is maximal if there is no ideal strictly between P and R. In a commutative ring, P is prime if and only if R/P is an integral domain, and P is maximal if and only if R/P is a field.
Ring Homomorphisms and Quotient Rings
A ring homomorphism f: R to S satisfies f(a+b) = f(a)+f(b) and f(ab) = f(a)f(b) for all a, b in R. If R is unital, we also require f(1 sub R) = 1 sub S. The kernel of f is an ideal of R, and the image is a subring of S. The quotient ring R/I has the same isomorphism theorems as for groups.
Canonical Example
The quotient ring Z/nZ is both a ring (quotient by the ideal nZ) and a group (quotient by the subgroup nZ). When n is prime, Z/nZ is a field. The quotient of the polynomial ring R[x] by the ideal generated by an irreducible polynomial p(x) yields a field extension of R.
7. Polynomial Rings, Irreducibility, and UFDs
Polynomial rings are among the most important rings in algebra. Understanding factorization and irreducibility in polynomial rings is essential for field theory and Galois theory.
Division Algorithm for Polynomials
If F is a field, then for any polynomials f(x) and g(x) in F[x] with g nonzero, there exist unique polynomials q(x) (quotient) and r(x) (remainder) in F[x] such that f(x) = q(x)g(x) + r(x) where the degree of r is less than the degree of g. This shows F[x] is a Euclidean domain and hence a PID.
Consequences of the Division Algorithm
- Factor theorem: (x - a) divides f(x) in F[x] if and only if f(a) = 0, i.e., a is a root of f.
- A polynomial of degree n over a field has at most n roots.
- Every ideal in F[x] is principal, generated by the gcd of its elements.
Irreducibility Tests
A polynomial is irreducible over a field F if it cannot be written as a product of two polynomials of smaller positive degree with coefficients in F. Irreducible polynomials are the primes of F[x].
Rational Root Test
If f(x) is a polynomial with integer coefficients and leading coefficient a sub n, then any rational root p/q (in lowest terms) must have p dividing the constant term and q dividing a sub n. This test quickly rules out rational roots, which suffices to prove irreducibility for degree 2 or 3 polynomials.
Eisenstein's Criterion
Let f(x) = a sub n times x to the n plus ... plus a sub 1 times x plus a sub 0 be a polynomial with integer coefficients. If there exists a prime p such that p divides a sub 0, a sub 1, ..., a sub (n-1) but p does not divide a sub n, and p squared does not divide a sub 0, then f is irreducible over the rationals. Classic application: x to the p minus 1 plus x to the (p-2) plus ... plus x plus 1 is irreducible over Q for any prime p.
Reduction Modulo a Prime
If reducing the coefficients of f modulo a prime p yields an irreducible polynomial over Z/pZ with the same degree, then f is irreducible over the rationals. This method works when Eisenstein does not apply directly.
Unique Factorization Domains
A unique factorization domain (UFD) is an integral domain in which every nonzero non-unit element factors uniquely as a product of irreducibles (up to order and units). The integers Z and polynomial rings F[x] over a field are both UFDs. Gauss's lemma shows that if R is a UFD, then R[x] is also a UFD.
8. Field Extensions and Algebraic Elements
Field theory studies field extensions — pairs E over F where E is a field containing F as a subfield. The theory connects algebra, geometry, and number theory through the notion of algebraic elements.
Algebraic vs. Transcendental Elements
An element alpha in an extension E of F is algebraic over F if it is a root of some nonzero polynomial with coefficients in F. Otherwise, alpha is transcendental over F. The element sqrt(2) is algebraic over Q (root of x squared minus 2), while pi and e are transcendental over Q.
Minimal Polynomial
If alpha is algebraic over F, its minimal polynomial is the monic polynomial of smallest degree in F[x] with alpha as a root. The minimal polynomial is always irreducible over F. The simple extension F(alpha) is isomorphic to F[x] modulo the ideal generated by the minimal polynomial, and the degree [F(alpha):F] equals the degree of the minimal polynomial.
Degree of a Field Extension
If E is an extension of F, then E is a vector space over F. The dimension of this vector space is called the degree [E:F]. The tower law states that if F is contained in K which is contained in E, then [E:F] = [E:K] times [K:F]. An extension of finite degree is called a finite extension, and every finite extension is algebraic.
Splitting Fields and Algebraic Closure
The splitting field of a polynomial f over F is the smallest extension of F over which f factors completely into linear factors. Splitting fields always exist and are unique up to isomorphism. The algebraic closure of F (written F-bar) is an algebraic extension in which every polynomial over F splits into linear factors. The algebraic closure is unique up to isomorphism and is algebraically closed: every nonconstant polynomial over F-bar has a root in F-bar.
Example: Splitting Field of x cubed minus 2 over Q
The polynomial x cubed minus 2 has roots: the real cube root of 2, and the real cube root of 2 times a primitive cube root of unity omega, and the real cube root of 2 times omega squared. The splitting field is Q(cube root of 2, omega), which has degree 6 over Q. This splitting field is a Galois extension of Q with Galois group isomorphic to S sub 3, the symmetric group on 3 elements.
9. Galois Theory: The Fundamental Theorem and Solvability
Galois theory, developed by Evariste Galois in the early 19th century, creates a remarkable dictionary between field extensions and group theory. It answers questions about which polynomial equations can be solved by radicals.
Galois Extensions and the Galois Group
A finite field extension E over F is Galois if it is both normal (every irreducible polynomial over F with one root in E splits completely in E) and separable (every element of E has a separable minimal polynomial). The Galois group Gal(E/F) is the group of all field automorphisms of E that fix every element of F. Its order equals the degree [E:F].
The Fundamental Theorem of Galois Theory
Let E/F be a finite Galois extension with Galois group G = Gal(E/F). There is an inclusion-reversing bijection:
Subfields K (F contained in K contained in E)
corresponds to
Subgroups H of G
- Given a subgroup H, the fixed field E to the H = elements of E fixed by all automorphisms in H.
- Given a subfield K, the subgroup Gal(E/K) consists of automorphisms of E fixing K pointwise.
- K is a normal extension of F if and only if Gal(E/K) is normal in G, and in that case Gal(K/F) is isomorphic to G/Gal(E/K).
Solvable Groups and Solvability by Radicals
A group G is solvable if there exists a chain of subgroups e = G sub 0 contained in G sub 1 contained in ... contained in G sub n = G where each G sub i is normal in G sub (i+1) and each quotient G sub (i+1) / G sub i is abelian. All abelian groups are solvable. The symmetric groups S sub 1, S sub 2, S sub 3, and S sub 4 are solvable, but S sub 5 and all S sub n for n at least 5 are not solvable.
Galois' Criterion for Solvability by Radicals
A polynomial f over a field F of characteristic 0 is solvable by radicals (its roots can be expressed using field operations and nth roots of elements of F) if and only if the Galois group of its splitting field over F is a solvable group.
The Abel-Ruffini Theorem
There is no general quintic formula.
The general polynomial of degree 5 has Galois group isomorphic to S sub 5 (over Q), which is not solvable because its only normal subgroup is A sub 5, which is simple and nonabelian. Therefore no formula involving radicals can express the roots of a general degree-5 polynomial. This contrasts with degrees 1 through 4, where solvable Galois groups guarantee the existence of radical formulas (the quadratic, cubic, and quartic formulas).
10. Modules, Exact Sequences, and Tensor Products
Modules generalize vector spaces: while a vector space has a field of scalars, a module has a ring of scalars. Module theory unifies the study of abelian groups, vector spaces, and representations.
Modules over a Ring
A left R-module M is an abelian group (M, +) together with a scalar multiplication R times M to M satisfying: r(m + n) = rm + rn, (r + s)m = rm + sm, (rs)m = r(sm), and 1m = m for all r, s in R and m, n in M. Submodules, module homomorphisms, quotient modules, and the isomorphism theorems all hold in direct analogy with groups and rings.
Free Modules
A free R-module of rank n is isomorphic to R to the n (the direct sum of n copies of R as a module over itself). Free modules have a basis, and every module is a quotient of a free module. Over a PID, every submodule of a free module is free.
Projective and Injective Modules
A projective module is a direct summand of a free module; equivalently, every surjective module map onto it splits. An injective module is one into which every injective map from a submodule extends to the ambient module. Over a field, all modules (vector spaces) are both projective and injective.
Exact Sequences
A sequence of module homomorphisms ... to A to B to C to ... is exact at B if the image of the map into B equals the kernel of the map out of B. Short exact sequences 0 to A to B to C to 0 express B as an extension of C by A. The sequence splits if B is isomorphic to the direct sum A plus C.
Tensor Products
The tensor product M tensor-over-R N of a right R-module M and a left R-module N is the abelian group generated by formal symbols m tensor n subject to bilinearity relations. The tensor product is characterized by the universal property that bilinear maps from M times N correspond bijectively to linear maps from M tensor N.
Key Tensor Product Identities
- R tensor-over-R M is isomorphic to M.
- Z/mZ tensor-over-Z Z/nZ is isomorphic to Z/gcd(m,n)Z.
- Tensor product distributes over direct sums: M tensor (A plus B) is isomorphic to (M tensor A) plus (M tensor B).
- For vector spaces over a field F, dim(V tensor W) = dim(V) times dim(W).
11. Applications: Coding Theory, Cryptography, and Physics
Abstract algebra is not purely theoretical. Its structures appear throughout applied mathematics, computer science, and physics.
Coding Theory: Linear Codes
A linear code is a subspace of the vector space (Z/2Z) to the n (or more generally F to the n for a finite field F). A (n, k) linear code encodes k bits of information into n-bit codewords, providing error detection and correction. The minimum distance of the code determines how many errors can be detected or corrected.
Cyclic Codes
A cyclic code is a linear code that is invariant under cyclic shifts of codewords. Cyclic codes correspond to ideals in the quotient ring (Z/2Z)[x] modulo (x to the n minus 1). Every cyclic code is generated by a single polynomial g(x) (the generator polynomial) that divides x to the n minus 1. Reed-Solomon codes, widely used in CDs, DVDs, and QR codes, are cyclic codes over finite fields. BCH codes (Bose-Chaudhuri-Hocquenghem) are another important family with well-understood error correction bounds.
Cryptography
Modern cryptography is built on algebraic structures. RSA encryption relies on the multiplicative group of units in Z/nZ (where n is a product of two large primes). Elliptic curve cryptography uses the group structure on points of an elliptic curve over a finite field, offering the same security as RSA with much smaller key sizes. The Diffie-Hellman key exchange relies on the computational hardness of the discrete logarithm problem in a cyclic group.
Physics: Symmetry Groups
Group theory is the language of symmetry in physics. Noether's theorem connects symmetries (group actions) to conserved quantities: translational symmetry gives conservation of momentum, rotational symmetry gives conservation of angular momentum, and time-translation symmetry gives conservation of energy.
Lie Groups in Physics
- SU(2) (special unitary group) describes spin in quantum mechanics and is the gauge group of the weak interaction.
- SU(3) is the gauge group of quantum chromodynamics (the strong nuclear force), with quarks transforming in the fundamental representation.
- The Standard Model gauge group is SU(3) times SU(2) times U(1).
- The Lorentz group SO(3,1) describes the symmetries of spacetime in special relativity.
12. Classification of Finite Simple Groups
Simple groups are the building blocks of all finite groups: a finite group is simple if it has no proper nontrivial normal subgroups. The complete classification of all finite simple groups, completed around 2004, is one of the greatest achievements in mathematical history — the proof spans tens of thousands of pages across hundreds of papers.
The Classification Theorem
Every finite simple group belongs to one of the following families:
Cyclic Groups of Prime Order
The cyclic group Z/pZ for any prime p. These are the abelian simple groups, and they are the only abelian simple groups.
Alternating Groups A sub n (n at least 5)
The alternating group A sub n is simple for every n at least 5. A sub 5, the smallest nonabelian simple group, has order 60. A sub 5 is also isomorphic to the icosahedral symmetry group and to PSL(2, 5), providing an early example of coincidences among simple groups.
Groups of Lie Type
These are groups that arise as matrix groups over finite fields, analogous to Lie groups over the real or complex numbers. They include the projective special linear groups PSL(n, q), the symplectic groups, the orthogonal groups, the unitary groups, and exceptional groups of types G2, F4, E6, E7, E8, and their twisted variants. Together these form several infinite families parameterized by a prime power q and a rank.
The 26 Sporadic Groups
Twenty-six simple groups do not belong to any infinite family. The largest is the Monster group M (also called the Friendly Giant), with order approximately 8 times 10 to the 53. The Monster has deep connections to number theory through monstrous moonshine, an unexpected relationship between the Monster and the coefficients of the modular j-function. The smallest sporadic group is the Mathieu group M sub 11 of order 7920.
Why Classification Matters
Just as prime numbers are the multiplicative building blocks of integers, simple groups are the group-theoretic building blocks of all finite groups (via composition series and the Jordan-Holder theorem). Knowing all simple groups is the first step in understanding all finite groups.
13. Practice Problems with Solutions
Work through these problems to test your understanding. Click to reveal each solution.
Problem 1: Identifying Non-Abelian Groups
Prove that the dihedral group D sub 4 (symmetries of a square) is non-abelian by exhibiting two elements that do not commute. What is the order of D sub 4?
Show Solution
D sub 4 has order 8 (4 rotations and 4 reflections). Label the vertices of the square 1, 2, 3, 4. Let r be rotation by 90 degrees (sending 1 to 2, 2 to 3, 3 to 4, 4 to 1) and let s be reflection across the horizontal axis of symmetry (sending 1 to 4, 2 to 3, 3 to 2, 4 to 1).
Computing rs: r sends 1 to 2, then s sends 2 to 3; r sends 2 to 3, then s sends 3 to 2; r sends 3 to 4, then s sends 4 to 1; r sends 4 to 1, then s sends 1 to 4. So rs sends (1,2,3,4) to (3,2,1,4).
Computing sr: s sends 1 to 4, then r sends 4 to 1; s sends 2 to 3, then r sends 3 to 4; s sends 3 to 2, then r sends 2 to 3; s sends 4 to 1, then r sends 1 to 2. So sr sends (1,2,3,4) to (1,4,3,2). Since rs and sr are different permutations, D sub 4 is non-abelian.
Problem 2: Applying Lagrange's Theorem
Let G be a group of order 35. Prove that G must be cyclic. (Hint: 35 = 5 times 7.)
Show Solution
By Sylow III, the number n sub 7 of Sylow 7-subgroups must divide 5 and satisfy n sub 7 congruent to 1 mod 7. The only divisor of 5 that is congruent to 1 mod 7 is 1, so n sub 7 = 1. Let P be the unique Sylow 7-subgroup; by Sylow II it is normal in G.
Similarly, n sub 5 must divide 7 and be congruent to 1 mod 5. The divisors of 7 are 1 and 7, and 7 is not congruent to 1 mod 5 (since 7 = 5 + 2 and 2 is not 0 mod 5), so n sub 5 = 1. Let Q be the unique Sylow 5-subgroup; it is also normal.
Since |P| = 7 and |Q| = 5 are coprime primes, P intersect Q = e. Since both are normal and their intersection is trivial, G is isomorphic to the direct product P times Q, which is Z/7Z times Z/5Z, which is isomorphic to Z/35Z. Therefore G is cyclic of order 35.
Problem 3: The First Isomorphism Theorem
Define f: Z to Z/6Z by f(n) = n mod 6. Identify the kernel, image, and the quotient group guaranteed by the first isomorphism theorem.
Show Solution
The map f is a ring (and group) homomorphism from (Z, +) to (Z/6Z, +). It is surjective since every residue class 0, 1, 2, 3, 4, 5 is hit. The kernel is the set of all integers n with n congruent to 0 mod 6, which is 6Z (the multiples of 6).
The first isomorphism theorem gives Z / (6Z) isomorphic to the image of f, which is all of Z/6Z. This is just the standard isomorphism Z/6Z isomorphic to Z/6Z, but it confirms that 6Z is a normal subgroup of Z (all subgroups of an abelian group are normal) and that the quotient has order 6.
Problem 4: Eisenstein's Criterion
Prove that f(x) = x to the 4 plus 4x plus 2 is irreducible over Q.
Show Solution
Apply Eisenstein's criterion with p = 2. Check the conditions:
- Leading coefficient: 1. The prime 2 does NOT divide 1. (Condition satisfied.)
- Non-leading coefficients: x to the 3 coefficient is 0 (divisible by 2), x squared coefficient is 0 (divisible by 2), x coefficient is 4 (divisible by 2), and constant term is 2 (divisible by 2). (All conditions satisfied.)
- Constant term squared check: 2 squared = 4 does NOT divide 2. (Condition satisfied.)
All three conditions of Eisenstein's criterion are met with p = 2, so f(x) = x to the 4 plus 4x plus 2 is irreducible over Q.
Problem 5: Galois Groups
Determine the Galois group of x to the 4 minus 5x squared plus 6 over Q. Is this polynomial solvable by radicals?
Show Solution
Factor the polynomial by substituting u = x squared: u squared minus 5u plus 6 = (u minus 2)(u minus 3). So x to the 4 minus 5x squared plus 6 = (x squared minus 2) (x squared minus 3). The roots are plus or minus sqrt(2) and plus or minus sqrt(3).
The splitting field is Q(sqrt(2), sqrt(3)), which has degree 4 over Q (since sqrt(3) is not in Q(sqrt(2))). The Galois group has order 4. The automorphisms are determined by their action on sqrt(2) and sqrt(3): each can be sent to its negative or left fixed, giving 4 automorphisms. The Galois group is Z/2Z times Z/2Z (the Klein four-group), which is abelian and hence solvable. Therefore the polynomial is solvable by radicals (which is obvious since the roots are square roots).
Problem 6: Prime and Maximal Ideals
In the ring Z[x], is the ideal (2) prime? Is it maximal? What about the ideal (2, x)?
Show Solution
The ideal (2) in Z[x]: the quotient ring Z[x]/(2) is isomorphic to (Z/2Z)[x], the polynomial ring over the field with 2 elements. Since Z/2Z is an integral domain (in fact a field), (Z/2Z)[x] is an integral domain, so (2) is a prime ideal. However (Z/2Z)[x] is not a field (x has no multiplicative inverse), so (2) is not maximal.
The ideal (2, x): the quotient Z[x]/(2, x) is isomorphic to Z/2Z (set both x = 0 and reduce mod 2). Since Z/2Z is a field, (2, x) is a maximal ideal. Every maximal ideal is prime, so (2, x) is also prime. Note that (2) is properly contained in (2, x), confirming that (2) is prime but not maximal.
Problem 7: Finitely Generated Abelian Groups
List all abelian groups of order 72 up to isomorphism, using the primary decomposition. (72 = 8 times 9 = 2 cubed times 3 squared.)
Show Solution
The abelian groups of order 72 are determined by partitions of the prime-power parts:
For the 2-primary part (order 8 = 2 cubed), the partitions of 3 are: 3 (giving Z/8Z), 2+1 (giving Z/4Z times Z/2Z), and 1+1+1 (giving Z/2Z times Z/2Z times Z/2Z).
For the 3-primary part (order 9 = 3 squared), the partitions of 2 are: 2 (giving Z/9Z) and 1+1 (giving Z/3Z times Z/3Z).
The abelian groups of order 72 are the 3 times 2 = 6 direct products, one from each choice:
- Z/8Z times Z/9Z (isomorphic to Z/72Z)
- Z/8Z times Z/3Z times Z/3Z
- Z/4Z times Z/2Z times Z/9Z
- Z/4Z times Z/2Z times Z/3Z times Z/3Z
- Z/2Z times Z/2Z times Z/2Z times Z/9Z
- Z/2Z times Z/2Z times Z/2Z times Z/3Z times Z/3Z
There are exactly 6 abelian groups of order 72 up to isomorphism.
Frequently Asked Questions
What are the four axioms that define a group?
A group requires closure (the operation stays in the set), associativity, an identity element, and inverses for every element. When commutativity also holds, the group is called abelian.
What does Lagrange's theorem say?
If H is a subgroup of a finite group G, then the order of H divides the order of G. The index [G:H] = |G|/|H| counts the number of left cosets of H in G.
What is the first isomorphism theorem?
If f: G to H is a group homomorphism, then G divided by the kernel of f is isomorphic to the image of f. This shows that every homomorphic image of G is a quotient of G.
What do the Sylow theorems tell us?
Sylow's theorems guarantee the existence of subgroups of prime power order, state that all such maximal subgroups are conjugate, and constrain the number of such subgroups. They are the main tools for classifying finite groups.
What is the Eisenstein irreducibility criterion?
A polynomial with integer coefficients is irreducible over the rationals if a prime p divides all non-leading coefficients, does not divide the leading coefficient, and p squared does not divide the constant term.
What is the fundamental theorem of Galois theory?
There is an inclusion-reversing correspondence between subfields of a Galois extension E/F and subgroups of the Galois group Gal(E/F). Normal subfields correspond to normal subgroups, and the Galois group of the subfield is the quotient.
Why is there no general formula for degree-5 polynomials?
The Abel-Ruffini theorem says the general quintic has Galois group isomorphic to S sub 5, which is not solvable. Galois' criterion states a polynomial is solvable by radicals if and only if its Galois group is solvable. Since S sub 5 is not solvable, no radical formula exists.
Quick Reference: Key Theorems
| Theorem | Statement (informal) | Area |
|---|---|---|
| Lagrange's Theorem | |H| divides |G| for any subgroup H of finite group G | Groups |
| First Isomorphism Theorem | G/ker(f) is isomorphic to im(f) for homomorphism f | Groups / Rings |
| Cauchy's Theorem | If prime p divides |G|, then G has an element of order p | Groups |
| Sylow Theorems | Existence, conjugacy, and count of Sylow p-subgroups | Groups |
| Cayley's Theorem | Every group embeds as a subgroup of some symmetric group | Groups |
| Jordan-Holder Theorem | Any two composition series have the same factors (up to order) | Groups |
| Eisenstein Criterion | Sufficient condition for irreducibility over Q via a prime | Rings |
| Gauss's Lemma | Product of primitive polynomials is primitive; R UFD implies R[x] UFD | Rings |
| Tower Law | [E:F] = [E:K] times [K:F] for tower F contained in K contained in E | Fields |
| Fundamental Theorem of Galois Theory | Subfields of E/F correspond to subgroups of Gal(E/F) | Fields |
| Abel-Ruffini Theorem | No radical formula exists for the general degree-5 polynomial | Fields |
| FTFGAG | Every finitely generated abelian group is a product of cyclic groups | Groups |
Further Study and Prerequisites
Prerequisites
- →Set theory and mathematical logic (injections, surjections, equivalence relations, partitions)
- →Linear algebra (vector spaces, matrices, determinants)
- →Number theory (divisibility, modular arithmetic, primes)
- →Mathematical proof techniques (induction, contradiction)
Advanced Topics
- →Representation theory: studying groups via their linear actions
- →Algebraic number theory: rings of algebraic integers, class groups
- →Homological algebra: derived functors, cohomology of groups
- →Category theory: abstract framework unifying all algebraic structures
- →Commutative algebra: localization, completion, Noetherian rings