## Week 1

### Introduction

• The origins of Group theory is in Premutations (the algebra obeyed) and Geometry (rotations) corresponding to Discrete and Continuous groups respectively.
• Geometric rotations, in general do not commute.
• Continuous groups is essentially Trignomentry.
• In quantum mechanics, symmetry group substitutes for the geometry of shape and size.

### Algebraic preliminaries

#### Sets and maps

Mathematicians usually classify maps as:

• Surjective/onto: Range is completely covered.
• Injective/into: One to One, but need not exhaust range.
• Bijective: Both one-to-one and covers the whole range.

Algebraic Structure: There exists a binary relation with properties:

• commutatitive: $$a \circ b$$ = $$b \circ a$$
• associative: $$a \circ (b \circ c) = (a \circ b) \circ c$$
• When not associative, Jacobi identity is used as an alternative.
• identity element: $$a \circ e = a = e \circ a$$
• inverse: $$a \circ a^{-1} = e = a^{-1} \circ a$$

Homomorphism: Maps which preserves algebraic structure.

Isomorphism: Maps which are bijective and preserves the algrbraic structure i.e., a map M: $$S_1 \rightarrow S_2$$ with algrbraic structures $$a \circ_1 b \in S_1$$ and $$a’ \circ_2 b’ \in S_2$$ has the property - $$(a \circ_1 b) \rightarrow (a’ \circ_2 b’)$$ if $$a \rightarrow a’$$ and $$b \rightarrow b’$$.

#### Groups

Group: A set which satisfies:

1. Closure: $$\forall a, b \in S\ \exists\ a \circ b \in S$$
2. Associative: $$a \circ (b \circ c) = (a \circ b) \circ c$$
3. Identity: $$\exists\ e \in S$$ such that $$a \circ e = a = e \circ a$$
4. Inverse: $$\forall a \in S\ \exists\ a^{-1} \in S$$ such that $$a \circ a^{-1} = e = a^{-1} \circ a$$

Examples of groups: set of matrices with determinant $$\ne$$ 0; set of all posible rotational configurations of a rigid object.

Abelian group: A group which satisfies an additional property - Commutativiy: $$a \circ b$$ = $$b \circ a$$. Example: Rotations in 2D plane - SO(2).

#### Linear vector space

A set $$V$$ with + (addition) and $$\bullet$$ (scalar multiplication) and an auxiliary set of scalars (s) which satisfies:

• Abelian group under +
• Under $$\bullet$$:
• $$a \bullet v \in V$$ when $$a \in s$$
• $$a \bullet (v_1 + v_2) = a \bullet v_1 + a \bullet v_2$$: multiplication is distributive
• Additionally, scalars have their own abelian structure with + so that $$a_1+a_2 \in s$$ etc.
• $$(a_1+a_2) \bullet v = a_1 \bullet v + a_2 \bullet v$$

#### Permutations

Permutation group is the group of all possible ways of rearranging n objects. The “possible ways” are elements of the group.

Any discrete group is a sub-group of some permutation group.

Can be represented as matrices.

#### Equivalence realtion

For a set $$s$$ a “relation” $$R$$ is a conditional about a,b etc. $$\in s$$ such that:

• $$a R a$$ : a is always related to a: Reflexivity
• $$a R b \Rightarrow b R a$$: Symmetry
• $$a R b$$ and $$b R c \Rightarrow a R c$$: Tansitivity

Any relation $$R$$ with above properties is called an Equivalence relation. Example: “parallel” relation of st. lines in a plane is an equivalence relation ; “perpendicular” is not an equivalence relation (does not satisfy first req.)

Theorem: An equivalence relation divides a set into disjoint subsets whose union makes up the whole set $$s$$.

Proof: Let $s_1, s_2, …$ be some subsets. Let $$s_1$$ be such that all elements in it are related by R. Similarly, consider $$s_2$$. Now, $$s_1 \cap s_2$$ because $$a \in s_1$$ and $$b \in s_2$$ and hypothesis $$a R b$$ forces $$s_1$$ and $$s_2$$ to be same subsets due to transitivity and symmetry.

## Week 2

### Lagrange’s Theorem:

If H is a subgroup

## Reference books:

• Morton Hammermesh
• Sadri Hassani Ch. 23 and 24
• Brian C. Hall
• Ramadevi’s draft book for applications