Lecture 01

Definition: Group

A group is a set $G$ equipped with a binary operation

\circ : G \times G \to G

s.t.

Associativity
- $(a \circ b) \circ c = a \circ (b \circ c)$
Existence of Identity:
- $\exists e \in G s.t. e \circ a = a \circ e = a \forall a \in G$
Existence of Inverse
- $\exists b \in G s.t. a \circ b = b \circ a = e$
- here $b$ is called the inverse of $a$ , written as $b = a^{- 1}$

$(Z, +) :$ inverse of $a$ is $- a$
$(Z_{p}, \cdot) :$ for any prime $p$
$(Q ∖ {0}, \cdot) :$ inverse of $\frac{a}{b}$ is $\frac{b}{a}$
$G L_{n} (C) :$ the General Linear Group is the set of $n \times n$ invertible matrices over the field of complex numbers, with matrix multiplication
$S L_{n} (C) :$ The Special Linear Group $S L_{n} \subseteq G L_{n}$ where $d e t (A) = 1 \forall A \in S L_{n}$
$D_{2 n} :$ Dihedral Groups

let $[n] := {1, 2, ..., n}$

and $π$ a bijection from $[n]$ to itself.

The set of all such bijections together with composition is called the Symmetric Group: $Sym_{n}$

Call $π$ a permutation.

π = (1 π_{1} 2 π_{2} \dots \dots n π_{n}) ⟺ π_{i} = π (i) 12 ⋮ n \to \to ⋱ \to π_{1} π_{2} ⋮ π_{n}

E.g.

π = (122331) ⟺ π_{i} = π (i) 123 \to \to \to 231

$a_{i} \in [n]$

$σ = (a_{1} a_{2} \dots a_{k})$ is a permutation s.t.
- $σ (a_{i}) = a_{i + 1}, \forall i = 1, \dots, k - 1$
- $σ (a_{k}) = a,$
- and for any other $a \neq = a_{i}, σ (a) = a$

$(123) (234) = (12) (34)$

because we read backwards

$2 \to 3 \to 1$

$4 \to 2 \to 3$

Any permutation $π \in Sym_{n}$ is a product of disjoint cycles

e.g.

(11243547536276) = (1) (2476) (35)