The text book for the course: Abstract Algebra, Third Edition, by Dummit and Foote, which is available as a Wiley Student Edition. The lectures may not always adhere to the text.
The course meets in room 217 on Tuesdays and Thursdays during 900-1100 hrs.
Evaluation will be as follows: continuous assessment (quizzes, attendance, assignments): 25%, mid-term: 30%, final: 45%.
While attendance is not compulsory, it is highly recommended that students regularly attend the lectures.

Syllabus (draft) pdf file

Videos: Some of the lectures are being recorded. Videos of those that are maybe found here.

Detailed list of topics follows.

Nilpotent and solvable groups:

• Review of group actions: class equation, proof of Sylow's theorems (pdf file of notes for this material); statement (without proof) of the simplicity of PSL_n(F_q) for n>1 except in the following three cases: n=2, q=2; n=2, q=3; and n=3, q=2; rough statement (without proof) of the list of finite simple groups
• p-groups; nilpotent groups; equivalent conditions for a finite group to be nilpotent: structure theorem for finite nilpotent groups; Frattini argument: a finite group is nilpotent if and only if every maximal (proper) subgroup of it is normal (DF 6.1)
• solvable and super-solvable groups; statements (without proofs) of Burnside's "p-a-q-b" theorem, Hall's characterization of solvable groups by existence of Sylow complements, Feit-Thompson theorem that odd order groups are solvable, and Thompson's theorem that a finite group is solvable if and only if every two generated subgroup of it is so. (DF 6.1, but definition of super solvability is not to be found there)

Homework due 23rd Jan: Numbers refer to exercises at the end of DF 6.1: 5, 16, 18, 19, 20, 22 (assuming that G is finite or, more generally, that every proper subgroup of N is contained in a maximal subgroup), 24, 25, 28; compute LCS and UCS of dihedral groups.

Elements of linear representations of groups:

• Representations of groups, equivariant morphisms between representations; equivalence with the category of modules over the group ring. (DF 18.1)
• Linearizing the action on a set: if G acts on X then it acts linearly on the free vector space FX over a field F; the left regular, right regular, and conjugation actions of a group G on FG; the notion of equivalence of linear representations. (DF 18.1)
• New actions (linear or otherwise) from old ones: if G acts on some object or a set of objects, it also acts naturally on anything constructed naturally from that set of objects; if X and Y are G-sets, then naturally so are X^n (cartesian power), 2^X (the power set of X), X^Y (functions from Y to X), etc.; if V and W are linear representations, then naturally so are tensor, symmetric, and exterior powers of V, the dual of V, Hom(V,W), combinations thereof, etc; the natural injection (which is an isomorphism when V is finite dimensional) of V^*\tensor W into Hom(V,W) is G-linear. (DF 18.1)
• Invariant subspace; irreducible or simple module; G-complements; for examples of subrepresentations not admitting G-complements in positive characteristic for a finite group, see converse of Maschke's theorem below; examples of such subrepresentations over the complex numbers (for necessarily infinite groups). (DF 18.1)
• Averaging process (assumption: G finite, charecteristic of the base field is coprime to |G|): projection onto the space of fixed points; Maschke's theorem: existence of G-complements; Converse of Mascke's theorem: if the characteristic of the field divides |G|, then the kernel of the map from the left regular representation FG to the trivial representation F sending all elements of G to 1 does not admit a G-complement; average of an inner product: unitarizability of a complex linear representation of a finite group.
• Notion of complete reducibility or semisimplicity; equivalent conditions for semisimplicity (of a module over an algebra over a field, the module being assumed finite dimensional over the field); quotients and submodules of semisimple modules are semisimple; notion of semisimplicity for a finite dimensional algebra over a field.
• Schur's lemma: the ring of G-endomorphisms of an irrep is a division ring; over an algebraically closed field, the only G-endomorphisms of an irrep are scalar multiplications. Corollary: any irrep of an abelian group over an algebraically closed field is 1-dimensional.

Homework to be completed 13th Feb (this will not be collected): All exercises at the end of DF 18.1.

Theory of characters:

• Character theory (some notes): the characters of irreps form an orthonormal basis for the Hilbert space of (complex valued) class functions on a finite group (with inner product (f,g):= reciprocal of |G| times the sum over x in G of the product of the conjugate of f(x) with g(x)). Corollaries (about complex finite dimensional reps of finite groups):
  • The multiplicity of an irrep in a representation is given by the inner product of their characters.
  • A representation is determined (up to isomorphism) by its character.
  • A representation is irreducible if and only if the inner product of its character with itself is 1.
  • The number of irreps equals the number of conjugacy classes of the group.
• Character tables: some examples; orthonormality of the rows hence also of the columns of the "weighted" character table: given two elements g and h of G, the sum over irreps V of X(g)X(h) is either |G|/C or 0 depending upon whether or not g and h are conjugates, where C denotes the cardinality of the conjugacy class of g (or that of h).
• Isotypical components; a formula for the G-projection onto an isotypical component.

Midterm on 26 February

Simple and semisimple finite dimensional algebras: notes

Elementary field theory:

• Fields; characteristic of integral domains; extension fields; degree of an extension field; elementary properties
• Simple extensions; adjoining roots; examples: cyclotomic extensions
• Finite and algebraic extensions; properties; irreducible polynomial of an algebraic element; algebraic closure in an extension; algebraically closed fields
• Frobenius morphism; perfect fields; examples: finite fields, fields of characteristic zero, algebraically closed fields; properties: algebraic extensions of perfect fields are separable.
• Separability of a polynomial; derivative criterion for existence of repeated roots; separable and inseparable degrees
• Splitting fields; lifting of isomorphisms to splitting fields; existence and uniqueness of splitting fields; existence and uniqueness of the finite field (of cardinality a prime power)
• Norms and traces

Homework to be completed 25th March (this will not be collected): All exercises at the end of DF 13.2.

Galois theory:

• Automorphisms of a field fixing a base field; Aut(E/F) is at most [E:F] with equality if E is the splitting field of a separable polynomial; Galois extensions; Galois group
• Dedekind's lemma: linear independence of characters;
• Artin's lemma on the fixed field of a finite group of field automorphisms: the resulting extension is Galois with the finite group as Galois group
• Main theorem of Galois theory: Galois correspondence
• Primitive element theorem: a finite extension is simple if and only if it admits finitely many subextensions;
• More on finite fields: Gauss's formula for the number of monic irreducibles of a given degree over a finite field: proof using Moebius inversion

Elements of commutative rings:

• Maximal ideals and prime ideals; primes pull back to primes under homomorphisms.
• Existence, using Zorn's lemma, of maximal ideals in a commutative ring with 1≠0; variations of the proof: maximal ideals containing a given ideal; ideals maximal with respect to containing an ideal I and not intersecting a multiplicatively closed set S with S∩I=empty.
• Ideals maximal with respect to not intersecting a multiplicatively closed set are prime.
• Localization of a commutative ring at a multiplicatively closed set S (with 1∈S); universal property.
• Localization of a module at a multiplicatively closed set S: it is the module over the localized ring at S obtained by extension of scalars.
• Zariski topology: the spectrum functor; maximal spectrum; spectra of the ring of integers, polynomial rings in one variable with real and complex coefficients.
• Extensions and contractions of ideals under homomorphisms: a prime ideal is contracted from a prime if it is contracted from its extension; the inclusion preserving correspondence between the spectrum of the localization at S and the set of those primes not meeting S.
• Radicals; nilradical is the intersection of all prime ideals.
• Prime avoidance; comaximal ideals: Chinese remainder theorem.
• Jacobson radical; its elemental characterization: x belongs to it iff 1-xy is a unit for every element y of the ring; Nakayama's lemma: proof by the determinant trick.