Groups & Linear Algebra (SC220)
Groups
Lecture 2
❖ Isomorphic Groups
→ In graph theory, unlabelled graphs are used to study abstract
structure of graphs whereas labelled graphs are used to study
application of graphs.
→ Similar to vertices of a labelled graph, elements of a group can
also be represented by any random symbol.
→ For example, consider a set G having six elements and all
elements are animal names.
G = {Dog, Cat, Cow, Wolf, Bear, Lion}
→ Relation Table:
Dog
Cat
Cow
Wolf
Bear
Lion
Dog
Dog
Cat
Cow
Wolf
Bear
Lion
Cat
Cat
Cow
Lion
Dog
Wolf
Bear
Cow
Cow
Lion
Bear
Cat
Dog
Wolf
Wolf
Wolf
Dog
Cat
Bear
Lion
Cow
Bear
Bear
Wolf
Dog
Lion
Cow
Cat
Lion
Lion
Bear
Wolf
Cow
Cat
Dog
→Suppose, above relation table is formed on a relation a*b = c
where a,b,c ∊ G, means that if animal a breaks friendship with
animal b then it makes new friendship with animal c.
→ Observing above table, we can conclude that set G and binary
operation * qualifies as a group because the set and the operation
satisfies the following four properties:
⇒ Closure: Applying * operation between any two elements of
the set results in an element that is also present in the set.
Example, Cat * Lion = Bear (Bear is present in G).
⇒ Identity: Dog is the identity element of this group. Because,
∀a ∈ G, a ∗ Dog = Dog ∗ a = a. Example, Dog * Cat = Cat,
Lion * Dog = Lion, etc.
⇒ Inverse: Each element x ∈ G has an element x−1 such that
x∗x−1 = x−1∗x = e. Here, the identity element e is Dog. Example,
Cat * Wolf = Dog, Cow * Bear = Dog, Lion * Lion = Dog,
Dog * Dog = Dog. So inverse of Dog, Cat, Cow, Wolf, Bear, Lion
are Dog, Wolf, Bear, Cat, Cow, Lion, respectively.
⇒ Associativity: The binary operation * is associative when
applied on three or more elements. Example,
Cat * (Lion * Bear) = Cow is same as (Cat * Lion) * Bear = Cow.
→ With symbols in place of numbers it is easy to study relations
between elements and practical applications of the groups.
→ Let the name of this group be “A”.
Consider the group U(18):
The set of U(18) will consist of all elements relatively prime to 18.
So, G = {1,5,7,11,13,17}
Group Table:
U(19)
1
5
7
11
13
17
1
1
5
7
11
13
17
5
5
7
17
1
11
13
7
7
17
13
5
1
11
11
11
1
5
13
17
7
13
13
11
1
17
7
5
17
17
13
11
7
5
1
→ In this group table, if 1, 5, 7, 11, 13, 17 are replaced with Dog,
Cat, Cow, Wolf, Bear, Lion respectively, the group table will match
exactly with the group table of group A.
→ Although both groups were created in different ways and
represented by different symbols, structure of both groups is same.
Such groups having different symbols for set elements and
different logics for binary operation are called isomorphic groups if
their group structure is same.
→ Formally, A group isomorphism is a function between two
groups that sets up a one-to-one correspondence between the
elements of the groups in a way that respects the given group
operations.
→ Suppose there are two groups (G,*) and (H,๏). A group
isomorphism is a bijective function f : G → H such that for all u and
v in G holds f(u * v) = f(u) ๏ f(v) . The two groups (G,*) and (H,๏)
are isomorphic if there exists an isomorphism from one to the
other. Notation for isomorphism: (G,*) ≅ (H,๏).
❖ Properties of Groups
(1)
Unique identity:
Theorem: The identity of element in a group is unique
Proof : Suppose, e1 and e2 be two identity elements of group (G, *),
e1, e2 ∈ G and e1 ≠ e2.
As e1 is an identity element,
e1 ∗ e2 = e2 -----------(i)
As e2 is an identity element
e1 ∗ e2 = e1 -----------(ii)
Comparing (i) and (ii),
e1 = e2
As this contradicts the assumption of e1 ≠ e2, a group cannot have
two identities. Thus the identity element is unique in group (G, *)
(2)
Unique inverse:
Theorem: The inverse of an element in a group is unique.
Proof : Suppose Ǝx in G such that y1 and y2 are it’s inverses and
y1≠ y2.
So, x ∗ y1 = y1 ∗ x = e and x ∗ y2 = y2 ∗ x = e
By associativity property (y1 ∗ x) ∗ y2 = y1 ∗ (x ∗ y2) ,
(y1 ∗ x) ∗ y2 = e ∗ y2 = y2
---------(i)
y1 ∗ (x ∗ y2) = y1 ∗ e = y1
---------(ii)
Equation (i) and (ii) are not equal. It violates associativity property
and hence contradicts the assumption of y1≠ y.
Thus, the inverse of an element in a group is unique.
(3) Inverse of inverse is the original element
Theorem: In a group (G,*), ∀ a ∈ G, (a-1)-1 = a.
Proof: Since a-1 is the inverse of a, therefore
⇒ a ∗ a-1 = e = a-1 ∗ a
⇒ a-1 ∗ a = e = a ∗ a-1
If x*y = e and y*x = e. Then the inverse of x is y, inverse of y is x.
By comparing a with x and a-1 with y,
⇒ inverse of a-1 = a
⇒ (a-1)-1 = a
Thus, inverse of inverse is the original element.
(4) Inverse of the result of a group operation is the product of
the inverses of the individual elements in reverse order.
Theorem: In a group (G,*), ∀ a,b ∈ G, (a * b)-1 = b-1 * a-1.
Proof : Since a, b, a-1, b-1, a*b, b-1 *a-1 all are elements of group G,
By associative property,
(a * b) * (b-1 * a-1) ⇒ a * (b * b-1) * a-1
⇒ a * e * a-1
⇒ a * a-1
⇒e
------(i)
(b-1 * a-1) * (a * b) ⇒ b-1 * (a-1 * a) * b
⇒ b-1 * e * b
⇒ b-1 * b
⇒ e
-----(ii)
From (i) and (ii),
(a * b) * (b-1 * a-1) = e = (b-1 * a-1)* (a * b)
If x*y = e and y*x = e. Then the inverse of x is y, inverse of y is x.
By comparing (a * b) with x and (b-1 * a-1) with y,
(b-1 * a-1)1 is inverse of (a * b)
⇒ (a * b)-1 = b-1 * a-1
Thus, the inverse of the result of a group operation is the product
of the inverses of the individual elements in reverse order.
(5) Group equations have unique solutions.
Theorem: In a group (G,*), ∀ a,x,y ∈ G, a ∗ x = a ∗ y ⇔ x = y.
Proof: Given, a ∗ x = a ∗ y
x⇒e*x
⇒ (a-1 * a) * x
⇒ a-1 ∗ (a ∗ x)
(By associative property)
⇒ a-1 ∗ (a ∗ y)
( a ∗ x = a ∗ y given )
⇒ (a-1 ∗ a) ∗ y
(By associative property)
⇒ e∗y
⇒ y
x = y.
Hence group equations have unique solutions.
❖ Group with three elements:
→ Suppose there is a group consisting of three elements,
G = {a,b,e}
→ Group table:
e
a
b
e
e
a
b
a
a
b
e
b
b
e
a
→ Each element of the group should be present in each row and
each column. After filling group table for identity element e, only
one option is left for a * a.
→ a * a cannot be a because a is already present once in that row.
→ a * a cannot be e because it will force a * b to be b which will
violate the rule as b is already present in that column.
→ Thus there is only one option left for a * b that is b.
→ Hence there is only one possible group table for any group
having three elements. This means all groups having three
elements are isomorphic.
Order of group:
→ The order of a group is its number of elements. It may be finite
or infinite.
→ If the order of a group is a prime number, then such groups are
called cyclic groups.
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