- A group is defined as a pair
consisting of a set and a binary operation on such that the following properties hold: - There exists an identity element
such that - (Fraleigh 4.17) There is only one such identity element.
- For all
, there exists an inverse element such that - (Fraleigh 4.17) Every element has only one such inverse element.
- The binary operation is associative. That is
- There exists an identity element
- We can denote the group operation as
or . The product can be denoted as , or simply . We also have exponentiation which can be denoted as or .
Group Properties
- The identity element is unique.
- The inverse of each element is unique.
- The Cancellation Law. That is
we have that
-
Inverse of Products
-
An element
is idempotent if . (Fraleigh e4.31) A group has exactly one idempotent element. - In fact, this idempotent element is
.
- In fact, this idempotent element is
-
Multiplication defines a bijection. Let
, then there is a bijection that is defined as Similarly
that is defined as -
The following formulation for groups called the One Sided Axioms are equivalent to the definition of groups.
-
The binary operation is associative
-
There exists a left inverse for each element
such that -
There exists a left identity
such that
-
-
The order of a group pertains to the number of elements. This is denoted
. - Lagrange’s Theorem on Orders If
is a finite group, then
- Lagrange’s Theorem on Orders If
-
The order of a group element
pertains to the smallest positive integer such that or if no such exists. We denote this by .