COURSE SYLLABUS

INTRODUCTION TO ABSTRACT ALGEBRA, SPRING 2014

Preliminaries

0.1: Examples of sets. Functions between sets. Images and fibers of functions.

Injective, surjective, and bijective maps. Permutations.

Binary relations. Equivalence relations. Equivalence classes and partitions.

0.2: Greatest common divisor. Least common multiple. a · b = gcd(a, b)·lcm(a, b).

The division algorithm. The Euclidean algorithm.

The Euclidean algorithm and the greatest common divisor. gcd(a, b) is a Z-linear combi-nation of a and b.

Fundamental theorem of arithmetic. Prime factorization and the gcd and lcm.

The Euler totient function ϕ.

0.3: Congruence classes and Z/nZ.

Number of units in Z/nZ is ϕ(n).

Part I - GROUP THEORY

Chapter 1: Introduction to groups

1.1: Axioms of a group. Abelian groups.

Examples of groups: Z/nZ under addition, (Z/nZ)× under multiplication.

Uniqueness of the identity. Uniqueness of inverses. Inverses of inverse elements. Inversesof products. Generalized associative law. Right and left cancellation laws.

The order of an element. The multiplication table of a group.

1.2: Dihedral groups D2n. Generators and relations for dihedral groups. Group presenta-tions.

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2 INTRODUCTION TO ABSTRACT ALGEBRA, SPRING 2014

1.3: Symmetric groups Sn. Row notation of a permutation. Cycle notation of a permuta-tion.

Cycle decomposition: any permutation can be written as a product of disjoint cyclesthat is unique up to rearrangement and cyclic rotation of cycles. Algorithm for finding thecycle decomposition of a permutation. Disjoint cycles commute and therefore the order of apermutation is the least common multiple of the lengths of its cycles.

Additional topic (not in textbook): Presentation of symmetric group using braid relations.

1.4: Definition of a field F . The general linear group GLn(F ).

1.5: The quaternion group Q8.

1.6: Definition of a homomorphism. Definition of an isomorphism. Properties shared byisomorphic groups.

Example 1: log : (R+,×)→ (R,+) satisfies log(xy) = log(x)+log(y) and log(1) = 0.

Example 2: the map τ 7→ f ◦ τ ◦ f−1 defines an isomorphism SA → SB when f : A → Bis a bijection.

1.7: Group actions. The permutation representation G→ SA associated to a group action.

Examples: Sn acts on {1, 2, · · · , n}. D2n acts on {1, 2, · · · , n}.

For n = 3, the permutation representation of the action of D6 on {1, 2, 3} defines a iso-morphism D6 → S3.

Chapter 2: Subgroups

2.1: Definition of a subgroup. Subgroup criterion.

2.2: The center Z(G) of a group G. The centralizer CG(A) of a subset of a group. Thenormalizer NG(A) of a subset of a group.

Kernels and stabilizers of group actions.

Actions of G on its underlying set: left multiplication, right multiplication, conjugation.

Centers, centralizers, and normalizers as kernels and stabilizers of the conjugation action.

2.3: Definition of cyclic groups. Any two cyclic groups of the same order are isomorphic.Number of generators in a cyclic group Zn.

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Additional topic (not in the textbook): Connection between the number of generators inZn and the number of units in (Z/nZ, ·).

2.4: The subgroup generated by a subset of a group. Definition as intersection of allgroups containing subset. Definition as group formed by noncommutative words in elementsof subset. Proof of the equivalence of the two definitions.

The order of the subgroup generated by a subset is independent of the order of its gener-ators. Examples:

- The subgroup of GL2(R) generated by a =

(0 11 0

)and b =

(0 2

1/2 0

)has infinite

order, even though |a| = |b| = 2.

- The symmetric group Sn has order n! but is generated by the n-cycle (1 2 · · ·n) and thetransposition (1 2).

2.5: The lattice of subgroups of a group. Examples: Z12 and D8.

Chapter 3: Quotient groups and homomorphisms

3.1: The kernel of a homomorphism. The kernel is a subgroup. The kernel of a groupaction is the kernel of the associated permutation representation.

The set of fibers of a group homomorphism forms a group. The fibers of a group homo-morphism are the cosets of the kernel.

The set of cosets G/K of a subgroup K of G forms a group if and only if K is normal.The kernel of a homomorphism is always a normal subgroup.

Every normal subgroup K �G is the kernel of a homomorphism G→ G/K.

Examples of quotient groups: Z/nZ, D8/Z(D8) ∼= V4.

3.2: Lagrange’s theorem: the order of a subgroup of G divides the order of G.

Corollary 1: the order of an element of a group divides the order of the group.

Corollary 2: Any group of prime order is cyclic.

Partial converses to Lagrange’s theorem: Cauchy’s theorem, Sylow’s theorem.

For subgroups A ≤ G and B ≤ G, the set AB is a subgroup if and only if AB = BA. Thisis true if A normalizes B or if B normalizes A. The order of AB.

3.3: First isomorphism theorem. Lifting homomorphisms to quotient groups.

4 INTRODUCTION TO ABSTRACT ALGEBRA, SPRING 2014

Second (diamond) isomorphism theorem. Third isomorphism theorem. Fourth (lattice)isomorphism theorem.

Example: lattice isomorphism theorem and D8/Z(D8).

3.4: Cauchy’s theorem for abelian groups.

Definition of a simple group. Definition of composition series. Jordan-Holder theorem.

Proof of Jordan-Holder theorem: existence statement uses existence of a maximal normalsubgroup as induction step, uniqueness statement uses the diamond isomorphism theorem asinduction step.

Chapter 4: Group actions

4.1: Orbits of group actions. Orbits and partitions. Bijection between the orbit of anelement and the quotient of G by stabilizer subgroup.

Transitive group actions. Sn acts transitively on {1, 2, · · · , n}. Orbits of cyclic subgroupsof Sn and the cycle decomposition of a permutation.

4.2: Cayley’s theorem.

Chapter 5: Direct and semidirect products

5.1: Direct products of groups. Subgroups and quotient groups of direct products.

5.4: Internal direct products.

5.5: Generalization of internal direct products to the case when one subgroup is notnormal. Semidirect products.

Part II - RING THEORY

Chapter 7: Introduction to rings

7.1: Definition of a ring. Commutative rings. Division rings and fields.

Zero divisors. Units. Integral domains. Cancellation property for integral domains. Afinite integral domain is a field.

Subrings.

7.2: Polynomial rings. Matrix rings. Group rings.

COURSE SYLLABUS 5

7.3: Definition of a ring homomorphism. Definition of a ring isomorphism. The imageand kernel of a ring homomorphism form subrings.

The set of cosets of a subring (considered as an additive subgroup) forms a ring if and onlyif the subring is an ideal. Quotient rings and the first isomorphism theorem for rings.

Second isomorphism theorem for rings. Third isomorphism theorem for rings. Fourthisomorphism theorem for rings.

Sums and products of ideals.

7.4: The ideal generated by a subset. Principal ideals. Finitely generated ideals.

A commutative ring is a field if and only if it has no non-trivial ideals. Maximal idealsand fields. Prime ideals and integral domains.

Chapter 8: Euclidean domains, principal ideal domains, and unique factoriza-tion domains

8.1: Norms and Euclidean domains. The Euclidean algorithm and greatest common divi-sors.

Every ideal in a Euclidean domain is principal (it is generated by any element of minimalnorm). Associates and greatest common divisors.

8.2: Definition of a principal ideal domain. Generators of principal ideals and greatestcommon divisors. Associates and principal ideals.

Every prime ideal in a principal ideal domain is maximal.

Dedekind-Hasse norms and principal ideal domains.

8.3: Irreducible and prime elements. Any prime element in an integral domain is irre-ducible. Partial converse: any irreducible element in a principal ideal domain is prime.

Example where the full converse fails: 2 · 3 = (1 +√−5)(1−

√−5), which implies that 2

and 3 are not prime in Z[√−5]. The existence of irreducible elements which are not prime

can be explained by the non-uniqueness of factorizations.

Definition of a unique factorization domain. Any irreducible element in a unique factor-ization domain is prime (proof uses analogue of Euclid’s lemma). Greatest common divisorsin a unique factorization domain.

Theorem: Every principal ideal domain is a unique factorization domain.

Proof of theorem: Existence of factorization can be interpreted as existence of chain ofprincipal ideals. Any ascending chain of ideals in a principal ideal domain (or more generally,

6 INTRODUCTION TO ABSTRACT ALGEBRA, SPRING 2014

a Noetherian ring) must terminate. Uniqueness of factorization follows from analogue ofEuclid’s lemma.

Corollary 1: The fundamental theorem of arithmetic (Z is a unique factorization domain).

Corollary 2: Every principal ideal domain has a Dedekind-Hasse norm.

A prime p ∈ Z is reducible in the ring of Gaussian integers Z[i] if and only if it can berepresented as a sum of squares. Fermat’s theorem of sums of squares.

Z[√−5] and Dedekind domains.

Part III - MODULES AND VECTOR SPACES

Chapter 10: Introduction to module theory

10.1: Definition of a module. Left and right modules, unital modules.

Examples: A module over a field is a vector space. Any abelian group has a uniqueZ-module structure, so abelian groups and Z-modules are equivalent. A module over thepolynomial ring k[x] is equivalent to a k-vector space V together with a choice of lineartransformation T : V → V .

An ideal I ⊂ R is an R-module. The direct product Rn is an R-module.

Definition of a submodule. Submodule criterion.

Interpretation: a module is equivalent to a ring action on an abelian group. Just as agroup action

G× A→ A

determines a group homomorphism

G→ SA,

an R-module structure on an abelian group M

R×M →M

determines a ring homomorphism

R→ End(M)

where End(M) is the endomorphism ring of M considered as a Z-module (cf section 10.2).

10.2: Module homomorphisms. Module isomorphisms. The kernel and image of a modulehomomorphism are submodules.

Examples: R-module homomorphisms for R a field are linear transformations. Any grouphomomorphism between abelian groups is a Z-module homomorphism. For R = k[x], an R-module homomorphism f : M → N is a linear transformation which intertwines the actionof x on M and N . A module isomorphism f : M → N exists if and only if M and N have the

COURSE SYLLABUS 7

same dimensions as vector spaces and the actions of x on M and N define similar matrices(this provides the basis for the applications of module theory to linear algebra covered inChapter 12).

The set of module endomorphisms of an R-module M forms a ring EndR(M).

Quotient modules: any submodule defines a quotient module (unlike the case of subgroupsor subrings). The quotient map and the first isomorphism theorem for modules.

The sum of two submodules.

Second isomorphism theorem for modules. Third isomorphism theorem for modules.Fourth isomorphism theorem for modules.

10.3: The submodule generated by a subset. Cyclic and finitely generated submodules.

The direct product of modules. Internal direct sums.

Free modules and their universal property. Additional topic: finitely presented modules.

10.4:

Chapter 11: Vector spaces

11.1:

11.2:

11.3:

11.4:

11.5:

Chapter 12: Modules over principal ideal domains

12.1:

12.2:

12.3: