Algebra

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 semi-direct products

  • structures of special types of groups such as:

    • p-groups
    • 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

  • Cayley-Hamilton 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

  • Finite-dimensional 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, Prentice-Hall, 1971.

  • Thomas W. Hungerford, Algebra, Springer, 1974.

  • Roy Smith, Algebra Course Notes (843-1 through 845-3),

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 \(Z(G) = [G, G]\))
  • Week 2 (May 25): Finite Groups
    • Special types of groups, the symmetric group, p-groups
    • Series of groups, solvable, simple, nilpotent; Jordan-Holder theorem
    • Group actions, orbit-stabilizer, 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, p-adics?
  • Week 8 (July 6) Linear Algebra
    • Bases, Singularity and invertibility, determinant and trace, rank-nullity (all over arbitrary fields)
    • Eigenstuff, minimal and characteristic polynomials
    • Canonical forms (RCF, JCF), decomposition into \(F[x]\)-modules
    • Spectral theorem, Cayley-Hamilton
    • 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 16-17
  • Topics
    • See the UGA official study guide
    • Group theory: Sylow theorems, p-groups, 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 semi-simple Lie algebras and Lie groups.
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