SC-220
GROUPS & LINEAR ALGEBRA
Scribe - 1
Topics
Relations
 Definition
 Types of Relations
Function
 Definition
 Types of Functions
Binary Relation and Binary Operator
Group
 Attributes of Group
 Properties of Group
 Examples
 Abelian Group
Other Ways to Define Group
 Connectivity between Group and Graph
 Connectivity between Group and Automaton
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Before diving into what is groups lets recall few basics!
Relations
➢ Any subset of a cartesian product is called a relation.
➢ The first set(input) is called domain.
➢ The second set(output) is called co-domain.
➢ A function is a special class of relation.
Types of Relations:
1. Reflexive : ∀ 𝑎∈𝑆, (𝑎,𝑎)∈𝑅
2. Irreflexive : ∃ 𝑎∈𝑆, 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 (𝑎,𝑎) ∉𝑅
3. Symmetric : ∀ 𝑥,𝑦∈𝑆, [(𝑥,𝑦)∈𝑅 ∧ (𝑦,𝑥)∈𝑅] ⊕ [(𝑥,𝑦)∉𝑅 ∧ (𝑦,x)∉𝑅]
4. Antisymmetric : ∀ 𝑥,𝑦∈𝑆, [(𝑥,𝑦) ∈ 𝑅 ∧ (𝑦,𝑥) ∈ 𝑅]⇒𝑥=𝑦
5. Transitive : ∀𝑥,𝑦,𝑧∈𝑆, [(𝑥,𝑦)∈𝑅 ∧ (𝑦,𝑧)∈𝑅]⇒(𝑥,𝑧)∈𝑅
6. non-transitive : ∃ 𝑥,𝑦,𝑧∈𝑆, such that (𝑥,𝑦)∈𝑅 ∧ (𝑦,𝑧)∈𝑅 ∧ (𝑥,𝑧)∉𝑅.
7. Equivalence relation : Any relation which are reflexive, symmetric
and transitive then this relation is called the equivalence relation.
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Function
Definition: It is a relation in which all elements of the domain is
connected with the only one element of the co-domain.
F: X→Y
Total No. of function : n(X)n(Y)
Types of Functions:
1.
Injective ( one - one ) : Injective function is a function in which
for all x in the domain there is a unique y in the co-domain.
x ≠ y => F(x) ≠ F(y)
2.
Surjective ( onto ) : Surjective function is a function in which for
every element y in the codomain of function, there is at least
one element x in the domain of function such that f(x) = y. In
other word we can say that co-domain is equal to range.
3.
Bijective (one – one & onto ) : It is a type of function which
satisfies the condition of both the injective and surjective
function.
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Binary Relation
Definition: A binary relation R from set A to set B is a subset of the
Cartesian product A×B: R⊆A×B.
If R⊆A×B is a binary relation and (a , b)∈R, we say a is related
to b by R. It is denoted by a R b.
Example:
The relation “greater than”, denoted by >, on the set A={1,2,3}.
The Cartesian square of the set A is given by
A2={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}.
We find all pairs (a , b) where a>b. This yields relation
R = {(2,1),(3,1),(3,2)}.
Binary Operator
Definition: Binary operations on a set are calculations that combine
two elements of the set (called operands) to produce another element
of the same set.
The binary operations * on a non-empty set A are functions from A × A
to A. The binary operation, *: A × A → A. It is an operation of two
elements of the set whose domain and co-domain are in the same set.
Example:
Addition is a binary operation on real number. So, if we add two
operands which are real numbers a and b, the result will also be a
real number.
Hence, + : R x R → R, (a, b) → a + b is a binary operator.
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Group
Definition: A group is a pair consisting of a set G and a binary
operation * defined in all pairs of elements of the set(including pairs
with element repeated).
Attributes of group
❖ Groups are defined with two attributes.
1. An underlying set.
2. A binary operation.
❖ We commonly use symbol G for underlying set and * for binary
operation.
❖ The value of the binary operation is denoted by placing the
operator between the two operands.
❖ Consider a non-empty set G and a function f : G X G → G is called
a binary operation on G. If * is binary operation on G, then it may
be written as a*b. ( a, b ∈ G )
❖ So, group is a subset of G X G X G.
Properties of groups
1. Closure property
▪ ∀ a, b ∈ G, (a * b) ∈ G
▪ The result of applying the operation ∗ to any pair of elements
must result in an element also in the set.
2. Identity property
▪ ∃ e ∈ G, such that ∀ a ∈ G, (e * a) = (a * e) = a.
▪ So, e is called the identity element.
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3. Inverse property
▪ ∀ a ∈ G, ∃ a-1 ∈ G, such that a * a-1 = e.
▪ a-1 = inverse of a.
▪ e = identity element.
4. Associative property
▪ ∀ a, b, c ∈ G, (a * b) * c = a * (b * c)
So, if we have a set and an operation, and it satisfies all the
properties, then it is a Group.
Examples of groups
E.g., 1) Addition over integer, (I,+) where I is integer.
To qualify as a group, the set and the operation must satisfy the
following four properties:
1. Closure property :∀ a, b ∈ I ⇒ a + b ∈ I.
2. Associative property :∀ a, b, c ∈ I , (a + b) + c = a + (b +c).
3. Identity property :∀ a ∈ I , there exist an element e such that a + e = a (Here e=0).
4. Inverse property :∀ a ∈ I, there exist an element b such that a + b = e(Here b = a).
As (I,+) satisfies all the properties. Therefore, (I,+) is an example of
group.
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E.g., 2) (Using Composition table)
Here we apply multiplication mod 10. 1,3,7 and 9 are relatively prime
to 10. [ (G ,*10) where G = {1,3,7,9} ]
*:GxG→G
*10
1
3
7
9
1
1
3
7
9
3
3
9
1
7
7
7
1
9
3
9
9
7
3
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• Elements of all rows and columns should be distinct.
1. Closure property :We can see that all the answers in above table is in set G. So it
satisfies the closure property.
2. Identity property :If any row or column is exactly same as row1 or column1, then
the first element of that row or column will be the identity
element.
(Here row1 is exactly same as row2, so the first element of
row2 is identity element, i.e. e = 1).
1*1 = 1
3*1 = 3
7*1 = 7
9*1 = 9
So, for this group 1 is identity element
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3. Inverse property :To find inverse first we find places where result of operation of
two elements is the identity element e.
From table:
1*1=1
3*7=1
7*3=1
9*9=1
So, every element has an inverse.
1 is inverse of 1
3 is inverse of 7
7 is inverse of 3
9 is inverse of 9
4. Associative property :(We can’t find directly using composition table).
(3 * 7) * 9 = 1*9 = 9 – (1)
3 * (7* 9) = 3*3 = 9
– (2)
So, from equation (1) and (2) 3 * (7* 9) = (3* 7) * 9
As (G,*10) satisfies all the properties. Therefore, (G,*10) is an
example of group.
Abelian group :
Definition: It satisfies all the first four properties of group including
fifth property (i.e., commutative property).
Example: In (I,+) ∀ a, b ∈ I , a + b = b + a (commutative property) is
satisfied. Therefore, (I,+) is an abelian group.
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Connectivity between Graph and Group
 Graph is defined on 2 parameters V vertex set and E an edge set.
 Vertex set is an independent set and edge set is defined from the
vertex set and so it is a dependent set.
 Edge set is a binary relation over vertex set.
 Binary operation takes two operands of underlying set (value
taken from the set can be repeated) and for every element
performs a predefined operation which maps to a unique image
belonging to the same set.
 If n is the number of elements in a set than there are total
possible number of pairs is n2.
 For a graph mathematically can be represented as,
VXV→E
 In similar way for a group,
GXG→G
 In Graph adjacency matrix is filled with 0 and 1 to show the
connections of the graph.
 In Groups, the table representation defines the formation of the
group and is commonly used to define group.
 It is filled with the elements of the underlying set.
Connectivity between Group and Automaton
❖ In Automaton simply vertex set is replaced with finite states from
Q(set of finite states).
❖ Simply it is a labelled directed graph.
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