Introduction to Group Theory

Definition of a Group

First, some mathematical symbols you should know:

A group $$(G,\circ)$$ is a set $$G$$ with a binary operation (explained below) $$\circ$$ that satisfies the following four axioms (which you will be able to test on the next page):

  1. Closure: $$\forall a,b \in G, a \circ b \in G$$
  2. Associativity: $$\forall a,b,c \in G, (a \circ b) \circ c = a \circ (b \circ c)$$
  3. Identity Element: $$\exists e \in G: \forall a \in G, a \circ e = e \circ a = e$$
  4. Inverse Element: For each $$a \in G, \exists b \in G: a \circ b = b \circ a = e$$ where $$e$$ is identity

$$\mathbb{N}=$$ Natural Numbers $$(1,2,3,...)$$
$$\mathbb{Z}=$$ Integers $$(...,-2,-1,0,1,2,...)$$
$$\mathbb{Q}=$$ Rational Numbers (any number that can be expressed as a fraction $$\frac{a}{b}; a,b\in\mathbb{Z}$$
$$\mathbb{R}=$$ Real Numbers (any rational or irrational number; everything but complex numbers)
$$\mathbb{C}=$$ Complex Numbers (any number of the form $$x+iy; x,i\in\mathbb{R}; i=\sqrt{-1}$$

The definition of a binary operation $$\circ$$ on a set $$H$$ is a rule which assigns each ordered pair of elements in $$H$$ to exactly one element in $$H$$. Multiplication and Addition are binary operations under $$\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$$. Subtraction is a binary operator under $$\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$$, but not under $$\mathbb{N}$$ because if $$a>b, b-a$$ is not defined. The same is true of division in both $$\mathbb{N}$$ and $$\mathbb{Z}$$ (for $$a,b\in\mathbb{Z}$$ or $$\mathbb{N}, \frac{a}{b}$$ is not defined. A binary operator does not necessarily have to be only addition, substraction, multiplication, or division. It can also be something like the least common factor (e.g. $$7\circ2=1, 10\circ4=2$$ in $$\mathbb{N}$$ or the least common multiple (e.g. $$12\circ16=48$$ in $$\mathbb{N}$$). Also, $$a \circ b$$ could be defined by $$(a \times b) \pmod{10}$$ (e.g. $$7 \circ 4 = 8$$ in $$\mathbb{Z}$$