# 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
• 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
• 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.