Introduction to Group Theory
The theory of groups, an important part in present day mathematics, started early in nineteenth
century in connection with the solutions of algebraic equations. Originally a group was the set of all
permutations of the roots of an algebraic equation which has the property that combination of any two of
these permutations again belongs to the set. Later the idea was generalized to the concept of an abstract
group. An abstract group is essentially the study of a set with an operation defined on it. Group theory has
many useful applications both within and outside mathematics. Groups arise in a number of apparently
unconnected subjects. In fact they appear in crystallography and quantum mechanics in geometry and
topology, in analysis and algebra and even in biology. Before we start talking of a group it will be fruitful to
discuss the binary operation on a set because these are sets on whose elements algebraic operations can be
made. We can obtain a third element of the set by combining two elements of a set. It is not true always.
That is why this concept needs attention.
Binary Operations
The concept of binary operation on a set is a generalization of the standard operations like addition
and multiplication on the set of numbers. For instance we know that the operation of addition (+) gives for
ally two natural numbers
another natural number
, similarly the multiplication operation gives
for the pair
the number
in again. These types of operations arc found to exist in many other
sets. Thus we give the following definition.
Binary Operation:
A binary operation to be denoted by
on a non-empty set
of elements
in
a unique element
of .
Alternatively a binary operation “ ” on
is a mapping from
is a rule which associates to each pair
to
i.e.
where the image of
of
under “ ”, i.e.,
, is denoted by
.
Thus in simple language we may say that a binary operation on a set tells us how to combine any
two elements of the set to get a unique element, again of the same set.
If an operation “ ” is binary on a set , we say that is closed or closure property is satisfied in , with
respect to the operation “ ”.
Examples:
(1) Usual addition (+) is binary operation on , because if
then
as we know that
sum of two natural numbers is again a natural number. But the usual subtraction (-) is not binary operation
on N because if
their
then
may not belongs to
which does not belong to .
. For example if
(2) Usual addition (+) and usual subtraction (-) both are binary operations on
then
and
.
and
because if
(3) Union, intersection and difference arc binary operations on
, the power set of .
(4) Vector product is a binary operation on the Set of all 3-Dimensional Vectors but the dot product is not a
binary operation as the dot product is not a vector but a scalar.
Types of Binary Operations
1. Commutative Operation: A binary operation
over a set
is said to be commutative, if for every pair
of elements
,
Thus addition and multiplication are commutative binary operations for natural numbers whereas
subtraction and division are not commutative because, for
for every pair of natural numbers
For example
and
and
can not be true
and .
.
2. Associative Operation: A binary operation a on a set
is called associative if
for all
.
Evidently ordinary addition and multiplication are associative binary operations on the set of natural
numbers, integers, rational numbers and real numbers. However, if we define
then
,
and
Thus, the operation defined as above is not associative.
3. Distributive Operation: Let and
be two binary operations defined on a set
operation
is said to be left distributive with respect to operation if
for all
and is said to be right distributive with respect to
if
. Then the
Whenever the operation
respect .
for all
is left as well as right distributive, we simply say that
is distributive with
Identity and Inverse
Identity: A composition in a set
exists an element
such that
is said to admit of an identity if these
Moreover, the element , if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to .
Examples:
(1) If
, the set of real numbers then
(Zero) is an additive identity of
because
the set of natural numbers, has no identity element with respect to addition because
(2)
is the multiplicative identity of
as
Evidently is identity of multiplication for
numbers).
Inverse: An element
exists
(set of integers),
(set of rational numbers,
is said to have its inverse with respect to certain operation
such that
being the identity in
.
with respect to .
Such an element , usually denoted by
is called the inverse of . Thus
(set of real
if there
for
.
In the set of integers the inverse of an integer
with respect to ordinary addition operation is
set of non-zero rational numbers, the inverse of
the set.
with respect to multiplication is
and in the
which belongs to
Algebraic Structure
A non-empty set
structure. Thus if
structure.
together with at least one binary operation defined on it is called an algebraic
is a non-empty set and “ ” is a binary operation on
, then
is an algebraic
are all algebraic structures. Since addition and multiplication are both binary operations on the set
real numbers,
is an algebraic structure equipped with two operations.
Example: If the binary operation
show that
Solution:
on
the set of rational numbers is defined by
for every
is commutative and associative.
(1) “ ” is commutative in
(2) “ ” is associative in
because if
because if
, then
then
of
Definition of Group
An algebraic structure
, where is a non-empty set with a binary
operation “ ” defined on it is said to be a group; if the binary operation satisfies the following axioms
(called group axioms).
(G1) Closure Axiom.
is closed under the operation
, i.e.,
, for all
(G2) Associative Axiom. The binary operation is associative. i.e.,
.
(G3) Identity Axiom. There exists an element
such that
element is called the identity of “ ” in .
(G4) Inverse Axiom. Each element of possesses inverse, i.e., for each element
element
such that
The element
is then called the inverse of a with respect to “ ” and we write
element of
.
. The
, there exists an
. Thus
is an
such that
Examples:
o
The structures
and
are not groups i.e., the set of natural numbers considered
with the addition composition or the multiplication composition, does not form a group. For,
the postulate (G3) and (G4) in the former case, and(G4) in the latter case, are not satisfied.
o
The structure
is a group, i.e., the set of integers with the addition composition is a
group. This is so because addition in numbers is associative, the additive identity belongs
to , and the inverse of every element , viz.
belongs to . This is known as additive
group of integers.
The structure
, i.e., the set of integers with the multiplication composition does not
form a group, as the axiom (G4) is not satisfied.
o
The structures
are all groups, i.e., the sets of rational numbers, real
numbers, complex numbers, each with the additive composition, form a group. But the same
sets with the multiplication composition do not form a group, for the multiplicative inverse
of the number zero does not exist in any of them.
o
The structure
is a group, where
is the set of non-zero rational numbers. This is
so because the operation is associative, the multiplicative identity
belongs to
multiplicative inverse of every element in the set is
, which also belongs to
is known as the multiplicative group of non-zero rational.
, and the
. This
Obviously
and
are groups, where
and
non-zero real numbers and non-zero complex numbers.
are respectively the sets of
Abelian Group or Commutative Group
If the commutative law holds in a group, then such a group is called an Abelian group or
Commutative group. Thus the group
if
,
is said to be an Abelian group or commutative group
.
A group which is not Abelian is called a non-Abelian group. The group
under addition while the group
is called the group
is known as group under multiplication.
Examples:
o
The structure
is a group, i.e., the set of integers with the addition composition is a
group. This is so because addition in numbers is associative, the additive identity belongs
to , and the inverse of every element , viz.
belongs to . This is known as additive
Abelian group of integers.
o
The structures
are all groups, i.e., the sets of rational numbers, real
numbers, complex numbers, each with the additive composition, form an Abelian group. But
the same sets with the multiplication composition do not form a group, for the multiplicative
inverse of the number zero does not exist in any of them.
o
The structure
is an Abelian group, where
is the set of non-zero rational
numbers. This is so because the operation is associative, the multiplicative identity
belongs to
also belongs to
, and the multiplicative inverse of every element
in the set is
, which
. This is known as the multiplicative Abelian group of non-zero rational.
Obviously
and
are groups, where
and
non-zero real numbers and non-zero complex numbers.
are respectively the sets of
Examples of Group
Example 1:
Show that the set of all integers …,-4, -3, -2, -1, 0, 1, 2, 3, 4, ... is an infinite Abelian group with
respect to the operation of addition of integers.
Solution:
Let us test all the group axioms for Abelian group.
(G1) Closure Axiom. We know that the sum of any two integers is also an integer, i.e., for all
,
. Thus is closed with respect to addition.
(G2) Associative Axiom . Since the addition of integers is associative, the associative axiom is satisfied,
i.e., for
Such that
(G3) Existence of Identity. We know that
i.e.,
Hence, additive identity exists.
(G4) Existence of Inverse. If
, then
is the additive identity and
,
. Also,
Thus, every integer possesses additive inverse. Therefore
is a group with respect to addition.
Since addition of integers is a commutative operation, therefore
Hence
is an Abelian group. Also,
an Abelian group of infinite order.
contains an infinite number of elements. Therefore
Example 2:
Show that the set of all non-zero rational numbers with respect operation of multiplication is a
group.
Solution:
Let the given set be denoted by . Then by group axioms, we have
(G1) We know that the product of two non-zero rational numbers is also a non-zero rational number.
Therefore
is closed with respect to multiplication. Hence, closure axiom is satisfied.
(G2) We know for rational numbers.
for all
Hence, associative axiom is satisfied.
(G3) Since, the multiplicative identity is a rational number hence identity axiom is satisfied.
is
(G4) If
, then obviously,
. Also
so that
is the multiplicative inverse of . Thus inverse axiom is also satisfied. Hence
respect to multiplication.
Example 3:
Show that
Solution:
, the set of all non-zero complex numbers is a multiplicative group.
Let
Here
(G1) Closure Axiom. If
numbers
Since
(G2) Associative Axiom.
and
, for
is the set of all real numbers and
(G3) Identity Axiom.
. Therefore,
is the identity in
(G4) Inverse Axiom. Let
Hence
and
is a multiplicative group.
.
, then by definition of multiplication of complex
is closed under multiplication.
for
Where
is a group with
, then
.
.
Order of a Group
Finite and infinite Groups:
If a group contains a finite number of distinct elements, it is called finite group otherwise an infinite
group.
In other words, a group
infinite.
is said to be finite or infinite according as the underlying set
is finite or
Order of a Group:
The number of elements in a finite group is called the order of the group. An infinite group is said to
be of infinite order.
Note: It should be noted that the smallest group for a given composition is the set
identity element alone.
consisting of the
Example:
1.
,
2.
,
is the example of finite group with order 3.
is the another example of finite group with order 4.
Composition Table or Cayley Table
A binary operation in a finite set can completely be described by means of a table. This table is
known as composition table. The composition table helps us to verily most of the properties satisfied by the
binary operations. This table can be formed as follows:
(i) Write the elements of the set (which are finite in number) in a row as well as in a column.
(ii) Write the clement associated to the ordered pair
the column headed by
and
. Thus (
entry on the left)
at the intersection of the row headed by
(
entry on the top) = entry where the
column intersect.
For example, the composition table for the group
for the operation of addition is given
and
row
below:
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
5
2
2
3
4
5
6
3
3
4
5
6
7
4
4
5
6
7
8
In the above example, the first element of the first row in the body of the table, 0 is obtained by adding the
first element 0of head row and the first element 0 of the head column. Similarly the third element
of 4th row (5) is obtained by adding the third element 2 of the head row and the fourth element of the head
column and so on.
An operation represented by the composition table will be binary, if every
entry of the composition table belongs to the given set. It is to be noted that composition table contains all
possible combinations of two elements of the with respect to the operation.
Note:
(1) It should be noted that the elements of the set should be written in the same order both in top border and
left border of the table, while preparing the composition table.
(2) Generally a table which defines a binary operation “.” on a set is called multiplication table, when the
operation is “+” the table is called an addition fable.
Group Tables
The composition tables are useful in examining the following axioms in the manner explained
below:
Closure Property: If all the elements of the table belong to the set
(say) then
is closed under the
Composition a (say). If any of the elements of the table does not belong to the set, the set is not closed.
Existence of Identity: The element (in the vertical column) to the left of the row identical to the top row
(border row) is called an identity element in the with respect to operation “ ”.
Existence of Inverse: If we mark the identity elements in the table then the element at the top of the
column passing through the identity element is the inverse of the element in the extreme left of the row
passing through the identity element and vice versa.
Commutative: If the table is such that the entries in every row coincide with the corresponding entries in
the corresponding column i.e., the composition table is symmetrical about the principal or main diagonal,
the composition is said to have satisfied the commutative axiom otherwise it is not commutative.
The process will be clearer with the help of following illustrative examples.
Example 1:
Prove that the set of cube roots of unity is an Abelian finite group with respect to multiplication.
Solution:
The set of cube roots of unity is
. Let us form the composition table as given below.
(G1) Closure Axiom: Since each clement obtained in the table is a unique element of the given set
multiplication is a binary operation. Thus the closure axiom is satisfied.
,
(G2) Associative Axiom: The elements of arc all complex numbers and we know that multiplication of
complex number is always associative. Hence associative axiom is also satisfied.
(G3) Identity Axiom: Since row of the table is identical with the top border row of elements of the
set, (the element to the extreme left of this row) is the identity element in .
(G4) Inverse Axiom: The inverse of
are
and
(G5) Commutative Axiom: Multiplication is commutative
main diagonal are equal to each other.
The number of elements in
is 3. Hence
respectively.
in because the elements equidistant with the
is a finite group of order 3.
Example 2:
Prove that the
Solution:
is Abelian multiplicative finite group of order 4.
Let
. The following will be the composition table for
.
(G1) Closure Axiom: Since all the entries in the composition table are elements of the set
closed under the operation multiplication. Hence closure axiom is satisfied.
, the set
is
(G2) Associative Axiom: Multiplication for complex numbers is always associative.
(G3) Identity Axiom: Row 1 of the table is identical with that at the top border, hence the element
extreme left column heading row is the identity clement.
(G4) Inverse Axiom: Inverse of
inverse axiom is satisfied in .
is . Inverse of
is
. Inverse of is
and of
in the
is . Hence
(G5) Commutative Axiom: Since in the table the 1st row is identical with 1st column, 2nd row is identical
with the 2nd column, 3rd row identical with the 3rd column and 4th row is identical with the 4th column,
hence the multiplication in is commutative.
The number of elements in
is 4. Hence
is an Abelian finite group of 4 with respect to multiplication.
Properties of Group
1. The identity element of s group is unique.
2. The inverse of each element of a group is unique, i.e., in a group
every
, there is only element
such that
with operation
,
for
being the identity.
3. The inverse of
, then the inverse of
is , i.e.,
.
4. The inverse of the product of two elements of a group is the product of the inverse taken in the
inverse order i.e.
.
5. Cancellation laws holds in a group, i.e., if
then
6. If
are any elements of a group
(left cancellation law),
is a group with binary operation
equations
and
and if
and
,
(right cancellation law).
are any elements of
have unique solutions in
, then the linear
.
7. The left inverse of an element is also its right inverse, i.e.
.
Modulo System
In the railway time table it is of common experience that is fixed with the prevision of 24 hours day
and night. When we say that a particular train is arriving at 15 hours, it implies that the train will arrive
at 3 p.m. according to our watch. Thus all the timing starting from 12 to 23 hours correspond to one of 0, 1,
3,…, 11 O’clock as indicated in watches. In other words all integers from 12 to 23 one equivalent to one or
the other of integers 0, 1, 2, 3, …, 11 with modulo 12, in saying like this integers in question are divided
into 12 classes.
In the manner described above the integer could be divided into 2 classes, or 5 classes
or m (m being a positive integer) classes and then we would have written mod 2 ormod 5 or mod m. This
system of representing integers is called modulo system.
Addition modulo
Now here we going to discuss with a new type of addition which is known as “addition modulo m”
and written in the form
By definition we have
Where
by .
where
and
belongs to an integer and
is the least non-negative remainder when
For example,
when
is divisible by .
, since
is any fixed positive integer.
, i.e., the ordinary addition of
and , is divided
, i.e., is the least non-negative reminder
Thus to find
integral multiples of
When
and
, we add and in the ordinary way and then from the sum, we remove
in such a way that the reminder is either or a positive integer less than .
are two integer such that
is divisible by a fixed positive integer
, then we
. Which is read as “a is concurrent to b mod m”.
have
Thus,
since
if and only if
is divisible by
is divisible by ,
. For example
,
,
.
Multiplication modulo
Now here we are going to define another new type of multiplication which is known as
“multiplication modulo p” and it can be written as
positive integer.
, where
By Definition, we have
Where
is the least non-negative remainder when
For example
, since
and
are any integers and
is a fixed
,
, i.e. the ordinary product of
.
and , is divided by .