Lecture 8 - Completion of a metric space - I.
- Definition of Cauchy sequences. Convergent sequences are Cauchy. Examples.

- The idea of completion of a metric space.

- \((x_n) , (y_n) \) converge to same limit point iff \( d(x_n,y_n) \to 0 \).

- Definition of equivalence relation on the set of Cauchy sequences in a metric space.

**Reading Exercise: ** Chapter 2 of Simmon's book : Introduction to topology and modern analysis.

Lecture 7 - Closed and Open subsets.
- Definition of open subsets and closed subsets of a metric space and examples.

- Limit point of a subset of a metric space.

- Thm: Closed iff there is point outside the set which is a limit point.

- Set obtained by adding limit points to a subset \(A\) is the smallest closed subset containing \(A\) and is
called its closure.

**Reading Exercise: ** Chapter 2 of Simmon's book : Introduction to topology and modern analysis.

Lecture 6 - Metric spaces.
- Definition of a metric space and examples : different metrics on \({\mathbb R}^n\), p-adic metric on rationals, etc.

- Definition of open ball in a metric. Examples in \({\mathbb R}^n\).

- Definition of convergent sequences.

**Reading Exercise: ** Chapter 2 of Simmon's book : Introduction to topology and modern analysis.

Lecture 5 - Insolvability of Quintic and work of Galois.
- Solvable groups.

- Fields, splitting fields of a polynomial (over \({\mathbb Q}\)) (intuitive idea).

- Brief overveiw of proof of insolvability of general quintic by Galois.

- Quotient group : Set of left cosets of a normal subgroup forms a group. Quotient map is a homomorphism. Example : \({\mathbb Z}/n\).

Lecture 4 - Normal subgroups, Quotient groups.
- Review of left cosets. Proof of Lagrange's theorem.

- Definition of a normal subgroup. Examples : trivial, whole, kernel of a homomorphism, subgroups of abelian groups.
Non-examples: upper triangular matrices in \({\mathsf GL}_n\), cyclic subgroups of \(S_5\).

- Right cosets. Left coset = right coset for a normal subgroup.

- Quotient group : Set of left cosets of a normal subgroup forms a group. Quotient map is a homomorphism. Example : \({\mathbb Z}/n\).

**Reading Exercise: ** Read Section 10 of Chapter 2 of Artin's algebra.

Lecture 3 - Homomorphisms, Cosets.
- Definition of homomorphism/automorphism. Examles : determinant, \({\mathbb Z}\xrightarrow{2}{\mathbb Z}\), trivial, identity, conjugation
- Kernel of a homomorphism/isomorphism. Proposition 5.13 of Artin: For a homomorphism \(\phi: G\to G'\), two elements \(a,b\in G\) go to same element of
\(G'\) iff \(a=bn\) for some \(n\in Ker(\phi)\).

- Left cosets : definition. Any two cosets are either disjoint or equal. Equivalence relation defining left cosets.

- The map from \(H \to aH\) defined by \(x \mapsto ax\) is bijective. Any two left cosets have same cardinality.

- Legrange's theorem: Order of a subgroup divides order of the group. (to be continued in next class)

**Reading Exercise: ** Read Section 4,5,6 of Chapter 2 of Artin's algebra.

Lecture 2 - Subgroups
- Review of the definition of groups. Cyclic groups, permutation groups.
- Cyclic subgroup generated by an element.

- Example: \({\mathsf GL}_n({\mathbb Z})\)

- Definition of a subgroup. Examples: trivial subgroup, whole subgroup, \( 2{\mathbb{Z}^+}\subset {\mathbb Z}^+, {\mathsf SL}_n({\mathbb R}) \subset {\mathsf GL}_n({\mathbb R}) \),
symmetries of a plane figure as subgroup of rigid plan motions.

**Reading Exercise: ** Read Section 2,3,4 of Chapter 2 of Artin's algebra.

Lecture 1 - Groups
- Examples of groups : \( {\mathbb Z}^+, {\mathbb R}^+, {\mathbb R}^{\times}, {\mathsf GL}_n({\mathbb R}), S_n, \) group of rigid motions of \({\mathbb R}^2, {\mathbb R}^3\)

- Definition of a binary operation or law of composition.

- Definition of a group.

- Order of an element.

**Reading Exercise: ** Read Section 1 of Chapter 2 of Artin's algebra.