Unit 3 | unit 3 group theory

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DMGT

Unit - 3

Group theory and ring theory

Q1) Define binary operation.

A1)

Binary operations-

A binary operation * is a set A is a function from A × A to A.

If * is a binary operation in a set A then than for the * image of the ordered pair (a, b) ∈ A×A, we write a*b.

For example:

Addition + is a binary operation in the set of natural number N, integer Z and real number R.

Multiplication is a binary operation in N, Q, Z, R and C.

Q2) What are the general properties of binary operations?

A2)

Propery-1: Let A be any set. A binary operation * A × A →A is said to be commutative if for every a, a, b ∈A.

a* b = b * a

Property-2: Let A be a non-empty set. A binary operation *; A × A →A is said to be associative if

a* b) * c = a * (b * c) for every a, b, c ∈A.

Property-3: Let * be a binary operation on a non-empty set A. If there exists an element e ∈A such that e * a = a * e = a for every a ∈A, then the element e is called identity with respect to * in A.

Property-4: Let * be a binary operation on a non-empty set A and e be the identity element in A with respect the operation *. An element a ∈A is said to be invertible if there exists an element b ∈A such that

a* b = b * a = e

In which case a and b are inverses of each other. For the operation * if b is the inverses of a ∈A then we can write b =

Q3) Define cancellation law.

A3)

A binary operation denoted by * in a set A, is said to satisfy.

(i) Left cancellation law if for all a, b, c ∈A,

a* b = a * c ⇒b = c

(ii) Right cancellation law if for all a, b, c ∈A

b* a = c * a ⇒b = c

Q4) Define group.

A4)

A group is an algebraic structure (G, *) in which the binary operation * on G satisfies the following conditions:

Condition-1: For all a, b, c, ∈ G

a* (b * c) = (a * b) * c (associativity)

Condition-2: There exists an elements e ∈G such that for any a ∈G

a* e= e * a = a (existence of identity)

Condition-3: For every a ∈G, there exists an element denoted by in G such that

a* = * a = e

is called the inverse of a in G.

Q5) What is Abelian group?

A5)

Let (G, *) be a group. If * is commutative that is

a* b = b * a for all a, b ∈G then (G, *) is called an Abelian group.

Q6) If G = {1, -1, i, -i} where i = , then show that G is an abelian group with respect to multiplication as a binary operation.

A6)

First we will construct a composition table-

.	1	-1	I	-i
1	1	-1	I	-i
-1	-1	1	-i	i
i	i	-i	-1	1
-i	-i	I	1	-1

It is clear from the above table that algebraic structure (G, .) is closed and satisfies the following conditions.

Associativity- For any three elements a, b, c ∈G (a ⋅b) ⋅c = a ⋅(b ⋅c)

Since

1 ⋅(−1 ⋅i) = 1 ⋅−i= −i

(1 ⋅−1) ⋅i= −1 ⋅i= −i

⇒1 ⋅(−1 ⋅i) = (1 ⋅−1) i

Similarly with any other three elements of G the properties holds.

∴ Associative law holds in (G, ⋅)

Existence of identity: 1 is the identity element (G, ⋅) such that 1 ⋅a = a = a ⋅1 ∀a ∈G

Existence of inverse: 1 ⋅1 = 1 = 1 ⋅1 ⇒1 is inverse of 1

(−1) ⋅(−1) = 1 = (−1) ⋅(−1) ⇒–1 is the inverse of (–1)

i⋅(−i) = 1 = −i⋅i⇒–iis the inverse of iin G.

−i⋅i= 1 = i⋅(−i) ⇒iis the inverse of –iin G.

Hence inverse of every element in G exists.

Thus all the axioms of a group are satisfied.

Commutativity: a ⋅b = b ⋅a ∀a, b ∈G hold in G

1 ⋅1 = 1 = 1 ⋅1, −1 ⋅1 = −1 = 1 ⋅−1

i⋅1 = i= 1 ⋅i; i⋅−i= −i⋅i= 1 = 1 etc.

Commutative law is satisfied.

Hence (G, ⋅) is an abelian group.

Q7) Prove that the set Z of all integers with binary operation * defined by a * b = a + b + 1 ∀a, b ∈G is an abelian group.

A7)

Sum of two integers is again an integer; therefore a +b ∈Z ∀a, b ∈Z

⇒a +b + 1 ⋅∈Z ∀a, b ∈Z

⇒Z is called with respect to *

Associative law for all a, b, a, b ∈G we have (a * b) * c = a * (b * c) as

(a* b) * c = (a + b + 1) * c

= a + b + 1 + c + 1

= a + b + c + 2

Also

a* (b * c) = a * (b + c + 1)

= a + b + c + 1 + 1

= a + b + c + 2

Hence (a * b) * c = a * (b * c) ∈a, b ∈Z.

Q8) Define subgroup.

A8)

Let (G, *) be a group and H, be a non-empty subset of G. If (H, *) is itself is a group, then (H, *) is called sub-group of (G, *).

Example-Let a = {1, –1, i, –i} and H = {1, –1}

G and H are groups with respect to the binary operation, multiplication.

H is a subset of G, therefore (H, X) is a sub-group (G, X).

Q9) If (G, *) is a group and H ≤G, then (H, *) is a sub-group of (G, *) if and only if

(i) a, b ∈H ⇒a * b ∈H;

(ii) a ∈ H ⇒∈H

A9)
If (H, *) is a sub-group of (G, *), then both the conditions are obviously satisfied.

We, therefore prove now that if conditions (i) and (ii) are satisfied then (H, *) is a sub-group of (G, *).

To prove that (H, *) is a sub-group of (G, *) all that we are required to prove is : * is associative in H and identity e ∈ H.

That * is associative in H follows from the fact that * is associative in G.

Also,

A ∈ H ⇒∈H by (ii) and e ∈H and ∈H ⇒a * = e ∈H by (i)

Hence, H is a sub-group of G.

Q10) What are the cosets?

A10)

Let (H, *) be a sub-group (G, *) and a ∈G

Then the sub-set:

a* H = {a * h: h ∈H }is called a left coset of H in G, and the subset

H * G = {h * a: h ∈H}is called a right coset of H in G.

Q11) If (H, *) is a sub-group (G, *), then a * H = H if and only if a ∈H. Prove

A11)

Let a * H = H

Since e ∈H then a = a * e ∈a * H

Hence a ∈H

Conversely

Let a ∈H then a * H ⊆H

(H, *) is a sub-group.

∴a ∈ H, h ∈ H ⇒ * h ∈ H.

Now h ∈H

⇒h = a * ( * h) ∈a * H

∴h ∈ H ⇒h ∈a * H

⇒H ⊆ a * H

Hence a * H = H

Q12) Define monoid.

A12)

Let S be a nonempty set with an operation. Then S is called a semigroup if the operation is associative. If the operation also has an identity element, then S is called a monoid.

(a) Consider the positive integers N. Then (N, +) and (N, ×) are semigroups since addition and multiplication on N are associative. In particular, (N, ×) is a monoid since it has the identity element 1. However, (N, +) is not a monoid since addition in N has no zero element.

(b) Let S be a finite set, and let F(S) be the collection of all functions f : S → S under the operation of composition of functions. Since the composition of functions is associative, F(S) is a semigroup. In fact,

F(S) is a monoid since the identity function is an identity element for F(S).

(c) Let S = {a, b, c, d}. The multiplication tables in Fig. B-1 define operations ∗ and ・ on S. Note that ∗ can be defined by the formula x ∗ y = x for any x and y in S. Hence

(x ∗ y) ∗ z = x ∗ z = x and x ∗ (y ∗ z) = x ∗ y = x

Therefore, ∗ is associative and hence (S, ∗) is a semigroup. On the other hand, ・ is not associative since,

For example,

(b ・ c) ・ c = a ・ c = c but b ・ (c ・ c) = b ・ a = b

Thus (S, ・) is not a semigroup.

Q13) What are the normal subgroups?

A13)

A subgroup H of G is a normal subgroup if Ha ⊆ H, for every a ∈ G, or, equivalently, if aH = Ha, i.e., if the right and left cosets coincide.

Note that every subgroup of an abelian group is normal.

Theorem:

Let H be a normal subgroup of a group G. Then the cosets of H form a group under coset multiplication:

(aH)(bH) = abH

This group is called the quotient group and is denoted by G/H.

Suppose the operation in G is addition or, in other words, G is written additively. Then the cosets of a subgroup H of G are of the form a +H. Moreover, if H is a normal subgroup of G, then the cosets form a group under coset addition, that is,

(a + H) + (b + H) = (a + b) + H

Q14) Define the kernel of homomorphism.

A14)

Let G and G be any two groups and f: G →be a homomorphism. Then Kernel of f denoted by Ker f the set K = (a ∈G: f (a) = ).

Where e is the identity of .

Q15) Let G be (Z, +) i.e., the group of integers under addition and let f: G → G defined by ∅(x) = 3x ∀x ∈G. Prove that f is homomorphism, determine its Kernel.

A15)

We have ∅(x) = 3x ∀x ∈G

∀x, y ∈G ⇒x + y ∈G (∴G is a group under addition)

Now

f(x + y) = 3 (x + y)

= 3x + 3y

= f (x) + f (y)

Hence f is homomorphism.

Kernel of homomorphism consists of half of zero i.e., the integers whose double is zero.

Thus K = {0}

Q16) Define ring.

A16)

A non-empty set R, equipped with two binary operations, called addition (+) and multiplication (.) is called a ring if the following postulates are satisfied.

(1) R is an additive Abelian group,

(2) R is an multiplicative semigroup

(3) The two distributive laws hold good, viz

a. (b + c) =a.b + a.c. (left distributive law)

(a + b). c = a.c + b.c, (right distributive law) for all a, b, c R