1. 2. Groups Basic Definitions Covering Operations Symmetric Groups
The Rearrangement Theorem
Isomorphism & Homomorphism
Subgroups
Cosets
Classes
Class Multiplication
Invariant Subgroups
Factor Groups
Direct Product Groups

2. 2.1. Basic Definitions Definition: Group
Let G be a set &  a multiplication.
{ G ,  } is a group if  a, b, c  G
1. a  b  G ( Closure )
2. a  ( b  c ) = ( a  b )  c ( Associativity )
3.  e  G  e  a = a  e = a ( Identity )
4.  a-1  G  a-1  a = a  a-1 = e ( Inverse )
Common abbreviations:
{ G ,  } → G
a  b → a b
Definition: Abelian Group
{ G ,  } is Abelian if a  b = b  a  a, b  G ( Commutative)
Common notations:  → + , e → 0, a-1 → –a

3. Definition: Order of G
Order of G = g = Number of elements in G
G is a finite (infinite) group if g is finite (infinite)
Examples:
G = { e, a } with a2 = e is a group of order 2 ( a-1 = a )
{ R, + } is an infinite Abelian group with e = 0 & a-1 = –a
{ R \ {0},  } is an infinite Abelian group with e = 1 & a-1 = 1/a
Definition: Order of a  G
an = e  Order of a = n
Definition: Cyclic Group Cn
Cn = { a, a2, …, an = e } is a cyclic group of order n

4. Multiplication Table
A (abstract) group can be defined by its multiplication table. a b  d a2 ab ad ba b2 bd   da db d2
Example: 4-group (Abelian)
V = { e, a, b, c = ab } V e a b c e a b c
MT of Abelian groups are symmetry
Dihedral group D2 = { e, C2x, C2y, C2z } has the same MT

5. 2.2. Covering Operations
Def: Covering operations of a geometric figure
= symmetry operations that leave the figure unchanged.
Example: Equilateral Triangle
The rotational symmetries form the group
Example: Equilateral Triangle
The covering operations form the group
 = reflections

6. C3v e C3
C32 1 2 3
Active rotation: