List of topics

I. Group theory

Constructions

• Direct product
• Semidirect product
• Quotient
• Set of cosets [set with group action]
• Free group
• Group defined by generators and relations

Examples

• Abelian groups [finite ones are canonically products of cyclic groups]
• Permutation groups
• Linear groups
• Automorphism groups [including Galois groups]
• Symmetry groups [especially dihedral groups]
• Multiplicative group of a ring [cyclic if the ring is a finite field]

Group actions

• Transitive iff looks like G/H [with translation action]
• Classification by orbits and stabilizers
• Fixed-point theorem for p-group actions
• Conjugation action (orbits, stabilizers, etc.)
• How to count (e.g., the class equation)

Analyzing the structure of a group

• Finding and using complements [i.e., recognizing semidirect products]
• Using Sylow's theorem
• Using information about G/Z(G)
• Constructing and using group actions

Simple groups

• Can't have all proper subgroups abelian
• The alternating groups are usually simple
• The projective special linear groups are usually simple

Theorems

• Cauchy and Sylow
• The 3 or 4 isomorphism laws
• A p-group has a non-trivial center
• p-groups have subgroups of all possible orders
• Finite cyclic groups have (unique) subgroups of all possible orders
• Subgroups of "small" index are normal

Universal mapping properties

• Quotients
• Free groups
• Groups defined by a presentation

II. Ring theory

Constructions and examples

• Direct product
• Polynomial ring
• Group ring
• Ring of algebraic integers
• Ring of functions
• Quotient
• Localization
• Universal root adjunction: F[x]/(f)
• Splitting field
• Finite fields

Ideals, etc.

• Prime ideals and maximal ideals
• prime elements, irreducible elements
• Gcd's exist in a UFD, related to ideals in the PID case
• All ideals are finitely generated iff the ring is noetherian

Actions, a.k.a. modules

• Cyclic modules and annihilators
• Recognizing direct sums; especially, splitting off a copy of R
• Free modules

Field extensions

• Algebraic vs. transcendental elements
• Mimimal polynomial
• Degree of an element, degree of a field extension
• Structure of a simple extension
• Separable extensions
• Normal extensions
• Galois extensions
• Galois group

Theorems

• Euclidean ==> PID ==> UFD ==> gcd's exist
• If R is a UFD, so is R[x]
• Hilbert basis theorem
• Chinese remainder theorem
• Zorn's lemma
• Over a noetherian ring, submodules of a finitely generated module are finitely generated
• Over a PID submodules of free modules are free, with a very nice form for the bases
• Structure theorem for modules over a PID
• Splitting fields are unique up to isomorphism and have lots of symmetry
• Primitive element theorem
• Galois correspondence between subgroups and subfields
• An equation is solvable by radicals iff its Galois group is solvable

Universal mapping properties

• Quotient rings and modules
• Polynomial rings
• Localization
• Root adjunction
• Free modules