Exploring Group Theory in Discrete Mathematics

Embark on a fascinating exploration of Group Theory in Discrete Mathematics through this enlightening PowerPoint presentation. This presentation delves into the foundational concepts of groups, providing a clear understanding of their structure, properties, and applications in various mathematical disciplines.

1. Abstract Algebra

2. Definition of a Group
A GroupGis a collection of elements
together with a binary operation* which
satisfies the following properties:
Closure
Associativity
Identity
Inverses
* A binary operationis a function on G
which assigns an element of Gto each
ordered pair of elements in G. For
example, multiplication and addition are
binaryoperations.
rubic cube permutation group
http://en.wikipedia.org/wiki/Permutation_group

3. Properties of a Group:
Closure [groupoid]
Example:
The Integers under Addition, (Z, +)
1 and 2 are elements of Z,
1+2 = 3, also an element of Z
Non-Examples:
The Odd Integers are not closed under
Addition. For example, 3 and 5 are odd
integers, but 3+5 = 8 and 8 is not an odd
integer.
The Integers lack inverses under
Multiplication, as do the Rational numbers
(because of 0.) However, if we remove 0 from
the Rational numbers, we obtain an infinite
closed group under multiplication.
“If we combine any two elements in the group under the binary
operation, the result is always another element in the group.” --Geoff
"members only"
http://en.wikipedia.org/wiki/index.html?curid=12686
870

4. Properties of a Group:
Associativity [semigroup]
The Associative Property,familiar from
ordinary arithmetic on real numbers,
states that(ab)c = a(bc). This may be
extended to as many elements as
necessary.
For example:
In Integers,
a+(b+c) = (a+b)+c.
In Matrix Multiplication,
(A*B)*C=A*(B*C).
In function composition,
f*(g*h)= (f*g)*h.
This is a property of allgroups.
Caution:
The Commutative Property,also familiar
from ordinary arithmetic on real numbers,
does not generally apply to all groups!
Only Abelian groups are commutative.
associative loop
http://en.wikipedia.org/wiki/List_of_algebraic_structures

5. Properties of a Group:
Identity [monoid]
The IdentityProperty, familiar from
ordinary arithmetic on real numbers,
states that, for all elements ain G,
a+e= e+a = a.
For example,
in Integers, a+0 = 0+a = a.
In (Q*, X), a*1 = 1*a = a.
In Matrix Multiplication, A*I=I*A=A.
This is a property of all groups.
The Identity is Unique!
There is only one identity
element in any group.
This property is used in
proofs.
|1 0|
|0 1|
= I

6. Properties of a Group:
Inverses
The inverseof an element, combined with that element, gives the identity.
Inverses are unique. That is, each element has exactly one inverse, and no two
distinct elements have the same inverse.
The uniqueness of inverses is used in proofs.
For example...
In (Z,+), the inverse of x is -x.
In (Q*, X), the inverse of x is
1
/
x
.
In abstract algebra, the inverse of an element ais usually written a
-1
.
This is why (GL,n)and (SL, n) do not include singular matrices; only nonsingular
matrices have inverses.
In Z
n
, the modular integers, the group operation is understood to be addition, because
if n is not prime, multiplicative inverses do not exist, or are not unique.
The U(n)groups are finite groups under modular multiplication.

7. Example
Thesetofallnxn
matricesunderthe
operationofmatrix
multiplicationisnota
groupsincenotevery
nxnmatrixhasits
multiplicativeinverse
butifGisthesetofall
nxnnon-singular
matrices,thenGforms
agroupunderthe
operationofmatrix
multiplication.

8. Abelian Groups
Abelian Groups are groups which have the
Commutative property, a*b=b*afor all aand bin G.
This is so familiar from ordinary arithmetic on Real
numbers, that students who are new to Abstract
Algebra must be careful not to assume that it
applies to the group on hand.
Abelian groups are named after Neils Abel, a
Norwegian mathematician.
Abelian groups may be recognized
by a diagonal symmetry in their
Cayley table (a table showing the
group elements and the results of
their composition under the group
binary operation.)
This symmetry may be used in
constructing a Cayley table, if we
know that the group is Abelian.
Neils Abel postage stamp http://en.wikipedia.org/wiki/Neils_Abel
Cayley tables for Z4 and U8
http://www.math.sunysb.edu/~joa/MAT313/hw-VIII---313.html

9. Subgroups
A subgroupHof a group Gis a subset of Gtogether
with the group operation, such thatHis also a group.
1.A semi group satisfies associative
2.A monoid is a semi group with an identity elements
3.A group is a monoid with Inverse property.
4.An Abelian group is a group with commutative
property.
5.A sub group inherits all the properties of group.
euler portrait
http://www.math.o
hio-
state.edu/~sinnott/
ReadingClassics/h
omepage.html

10. Consider binary relation * defined
on set A={A,B,C,D} by following
table and check commutative and
Associative
euler portrait
http://www.math.o
hio-
state.edu/~sinnott/
ReadingClassics/h
omepage.html
* A B C D
A A C B D
B D A B C
C C D A A
D D B A C
Commutative
Associative

11. Prove the set ß={0,1,2,3,4} is a
finite abelian group of order 5
under addition
euler portrait
http://www.math.o
hio-
state.edu/~sinnott/
ReadingClassics/h
omepage.html
+501234
0
1
2
3
4
Closure Property
Associative Property
Addition is an associative operation, and this property holds
for all elements in β.
Identity Property
The element 0 is the identity element because, for all a in β, 0
+ a = a + 0 = a. This fulfills the requirement for an identity
element.
Inverse Property
Commutative Property
0 + 0 = 0
0 + 1 = 1
0 + 2 = 2
0 + 3 = 3
0 + 4 = 4
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 0
2 + 2 = 4
2 + 3 = 0
2 + 4 = 1
3 + 3 = 1
3 + 4 = 2
4 + 4 = 3
•For 0, its inverse is 0.
•For 1, its inverse is 4 (1 + 4 = 4 + 1 = 0).
•For 2, its inverse is 3 (2 + 3 = 3 + 2 = 0).
•For 3, its inverse is 2 (3 + 2 = 2 + 3 = 0).
•For 4, its inverse is 1 (4 + 1 = 1 + 4 = 0).

12. Prove the set ß={0,1,2,3,4} is a
finite abelian group of order 5
under addition
euler portrait
http://www.math.o
hio-
state.edu/~sinnott/
ReadingClassics/h
omepage.html
+501234
0
1
2
3
4
Closure Property
Associative Property
Identity Property
Inverse Property
Commutative Property

13. Let Z={1,2,..n-1} let
euler portrait
http://www.math.o
hio-
state.edu/~sinnott/
ReadingClassics/h
omepage.html
*7123456
1
2
3
4
5
6
Closure Property
Associative Property
Identity Property
Inverse Property
Commutative Property