2.2 Groups

2.2 Groups

This section provides an overview of basic algebra objects and their properties.

Definition 2.2.1 A binary operation * on a set S is a mapping from S x S to S. That

is, * is a rule which assigns to each order pair of elements from S an element of S.

Definition 2.2.2 A group operation (G, *) consists of a set G with a binary operation *

on G satisfying the following three axioms.

1. The group is a associative. That is, a * (b * c)=(a * b) * c for all a, b, c Є G.

2. There is an element 1 Є G, called the identity element, such that a * 1=1* a = a

for all a Є G.

3. For each a Є G there exists an element a^(-1) Є G, called the inverse of a, such that

a * a^(-1) = a^(-1) * a =1.

A group G is abelian (or commutative) if, furthermore,

4. a * b = b * a for all a, b Є G.

Definition 2.2.3 A group G is a finite if |G| is finite. The number of elements in a finite

group is called its order.

Definition 2.2.4 A group G is a cyclic if there is an element g Є G such that for each

b Є G there is an integer i with b = g^i. Such an element g is called a generator of G.