Group - Group Lecture 14 Discrete Mathematic Structures...

Group
Lecture 14
Discrete Mathematic Structures

Group
•
Group axioms
–
Association
–
Identity
–
Inverse property
•
Example
–
Addition group on integers (Z,+)
–
All one-to-one functions on {1,2,3}, plus
composition of function: S
3

Inverse Property of a System
•
For any element in a system, there may or
may not be its inverse.
•
However, “for any element x in S , x has its
inverse” is a property of the system as a
whole.
•
For a system for which the inverse
property holds, each element has its
particular inverse.

Semigroup and Group
•
A group is a semigroup
–
Association
•
A group is a monoid
–
Identity
•
Negative exponential
–
Denotation of inverse of element a : a
-1
–
Expansion of exponential: a
-k
=(a
-1
)
k
(k is positive
integer)
•
Abelian group: commutative group

An Example of Abelian Group
•
Let G be the set of all nonzero real numbers, let
a * b = ab /2, then ( G ,*) is a Abelian group.
•
Verifying that all requirements as described as
definition are satisfied:
–
“*” is a closed binary operation on G .
–
Associability : ( a * b )* c = a *( b * c )=( abc )/4
–
Identity : a *2=2* a for all a in G
–
Inverse : examine the equation a * x = 2, it is easy to see
that for all a in G , a
–1
=4/ a
–
Commutability : obviously, a * b = b * a

Example:
•
Let (S,° ) be group, u is a special element of S,
define binary operation *: a*b = a° u
-1
° b. then
(S, *) is a group G
–
Associability G
•
(a*b)*c = a*(b*c) = a3 u -1 o b° u -1 G c
–
Identity G
•
G ° °x, x*u = x° u -1 ± u =x, G u*x = u° u -1 ± x = x
–
Inverse G
•
G ° °x, x*(u° x -1 ± u)=x3 u -1 ¨ (u‚ x -1 ± u) = u
x
-1
is the inverse if x in group (S, ° )

Some properties:
•
Each element in group has only one
inverse
•
G be a group, a,b,c be elements of G, then
–
ab=ac implies that b=c
–
ba=ca implies that b=c
•
G be a group, a,b be elements of G, then
–
(a
-1
)
-1
= a
–
(ab)
-1
= b
-1
a
-1

Group Equation and Its Solution
•
Group equations:
–
aSx = b and ySa = b , where a , b are constants
•
Solutions of the group equations:
–
aSx = b  aS ( a
-1
G b ) = b
–
ySa = b  ( bS a
-1
) G a = b
•
The group equation has unique solution:
– Assuming that aSx
1
= b = aSx
2
, multiply the two sides
of the equation from the left with a
-1
, x
1
= a
-1
G b = x
2
,

Second Definition of Group
•
( G ,e ) is an algebraic system, if association holds for it,
and the two equation aSx = b and ySa =b have unique
solutions each, then ( G ,° ) is a group.