Group Theory

Subgroups and quotient groups

Lagrange’s Theorem

Fundamental homomorphism theorems

Group actions with applications to the structure of groups such as the Sylow Theorems

group constructions such as direct and semidirect products

structures of special types of groups such as:
 pgroups
 dihedral,
 symmetric
 alternating

Cycle decompositions

The simplicity of \(A_n\), for n \(\geq\) 5

Free groups, generators and relations

Solvable groups
References: 1,3,4
Linear Algebra

Determinants

Eigenvalues and eigenvectors

CayleyHamilton Theorem

Canonical forms for matrices

Linear groups (\(\operatorname{GL}_n , {\operatorname{SL}}_n, O_n, U_n\))

Duality
 Dual spaces,
 Dual bases,
 Induced dual map,
 Double duals

Finitedimensional spectral theorem
References: 1,2,4
Foundations

Zorn’s Lemma and its uses in various existence theorems such as that of
 A basis for a vector space or
 Existence of maximal ideals.
References: 1,3,4
Rings and Modules

Basic properties of ideals and quotient rings

Fundamental homomorphism theorems for rings and modules

Characterizations and properties of special domains such as:

Euclidean implies PID implies UFD

Classification of finitely generated modules over PIDs with emphasis on Euclidean domains applications to the structure of:

Finitely generated abelian groups

Canonical forms of matrices

References: 1,3,4
Field Theory

Algebraic extensions of fields

Fundamental theorem of Galois theory

Properties of finite fields

Separable extensions

Computations of Galois groups of polynomials of small degree and cyclotomic polynomials solvability of polynomials by radicals
References: 1,3,4
As a general rule, students are responsible for knowing both the theory (proofs) and practical applications (e.g. how to find the Jordan or rational canonical form of a given matrix, or the Galois group of a given polynomial) of the topics mentioned.
References

David Dummit and Richard Foote, Abstract Algebra, Wiley, 2003.

Kenneth Hoffman and Ray Kunze, Linear Algebra, PrenticeHall, 1971.

Thomas W. Hungerford, Algebra, Springer, 1974.

Roy Smith, Algebra Course Notes (8431 through 8453),
Notion Page
Scheduling

Week 1 (May 18): Preliminary Review

Subgroups, quotients, isomorphism theorems, cosets, index of a subgroup
 One step subgroup test
 Cauchy, Lagrange
 Everything about: cyclic, symmetric, alternating, dihedral groups of order ≤ 20
 Especially useful problems: if G is nonabelian of order \(p^3\), then \(Z(G) = [G, G]\))

Subgroups, quotients, isomorphism theorems, cosets, index of a subgroup

Week 2 (May 25): Finite Groups
 Special types of groups, the symmetric group, pgroups
 Series of groups, solvable, simple, nilpotent; JordanHolder theorem
 Group actions, orbitstabilizer, class equation,
 Cayley representation, permutation representation

FTFGAG: The Fundamental Theorem of Finitely Generated Abelian Groups
 Invariant factors
 Elementary divisors
 How to go back and forth
 Recognition of direct products and semidirect products

Week 3 (June 1): Sylow Theorems
 Showing groups are abelian
 Classification: isomorphism classes of groups of a given order, recognizing direct and semidirect products

Week 4 (June 8): Rings and Commutative Algebra
 Morphisms, Ideals, quotients, zero divisors, isomorphism theorems, CRT
 Irreducible and prime elements, nilpotent, units
 Radical, nilradical, spec and maxspec
 Special types: domains, integral domains, Euclidean ⇒ PID ⇒ UFD ⇒?, Dedekind domains, Noetherian, Artinian
 Zorn’s lemma arguments
 Bonus optional stuff: localization

Week 5 (June 15): Modules and Homological Algebra
 Morphisms, submodules, isomorphism theorems, principal ideals
 Free and projective, free rank, torsion submodule, annihilators
 Tensor product
 SESs and splitting
 Flat and torsionfree
 Classification theorem for modules over a PID, elementary divisors and invariant factors
 \(M/IM\) stuff, \(R[x]\) modules

Week 6 (June 22): Field Theory
 Finite and algebraic extensions, degrees, towers
 Finite fields
 Separable and normal extensions, splitting fields
 Irreducible polynomials
 Algebraic closure
 Primitive element theorem, perfect fields, Frobenius

Week 7 (June 29): Galois Theory and Number Theory
 Fundamental theorem
 Computing Galois groups
 Solvability by radicals
 Cyclotomic polynomials, primitive roots
 Properties in towers (normal, separable, Galois, etc)
 Bonus: rings of integers, splitting of primes, padics?

Week 8 (July 6) Linear Algebra
 Bases, Singularity and invertibility, determinant and trace, ranknullity (all over arbitrary fields)
 Eigenstuff, minimal and characteristic polynomials
 Canonical forms (RCF, JCF), decomposition into \(F[x]\)modules
 Spectral theorem, CayleyHamilton
 Similarity, diagonalizability, simultaneously diagonalizable operators
 (Bonus) Quadratic and bilinear forms, lattices, Gram matrix, special types of matrices (Hermitian, symmetric, orthogonal, unitary, etc)

Week 9 (July 13): More linear algebra, or (Bonus) Representation Theory of Finite Groups
 Maschke’s theorem
 Orthogonality of characters, character tables
 Schur’s lemma
 Week 10 (July 20): Buffer
 Week 11 (July 27): Buffer
 Week 12 (August 3): No meeting (Mock AMS)
 Week 13 (August 10): Timed practice exam
 Quals: Monday and Tuesday August 1617

Topics
 See the UGA official study guide
 Group theory: Sylow theorems, pgroups, solvable groups, free groups.
 Rings and modules: tensor products, determinants, Jordan canonical form, PID’s, UFD’s, polynomials rings.
 Field theory: splitting fields, separable and inseparable extensions.
 Galois theory: Fundamental theorems of Galois theory, finite fields, cyclotomic fields.

Extra Topics
 Representations of Finite Groups: character theory, induced representations, structure of the group ring.
 Basics of Lie groups and Lie algebras: exponential map, nilpotent and semisimple Lie algebras and Lie groups.