Mathematical Foundation of Computer Science

DHARM

ALGEBRAIC STRUCTURES 99

EXERCISES

4.1 Let X = {0, 1, 2, 3, 4} then show that,

(i) Algebraic structure (X, + 5 ) is an Abelian group and

(ii) Algebraic structure (X, * 5 ) is also an Abelian group.

where binary operations ‘+ 5 ’ and ‘* 5 ’ are addition modulo 5 and multiplication modulo 5 respec-

tively.

4.2 Is the algebraic structure (X, * 5 ) forms an finite Abelian group where the set X = {1, 2, 3, 4, 5,

6}. Determine the order of the group.

4.3 Define homomorphism on a semigroup. Explain when a homomorphism becomes an isomor-

phism.

4.4 Define a cyclic group. Show that following groups are cyclic,

(i) Algebraic structure ({1, – 1, i, – i}, *) where i = – 1.

(ii) Algebraic structure Σ where ω is cube root of unity.

4.5 Show that group (C, *7) is a cyclic group, where set C = {1, 2, 3, 4, 5, 6}.

4.6 Define the order of an element in a group. Find the order of each element in the following

multiplicative group,

(i) ({1, – 1, i, – i}, *) (ii) ({1, ω, ω^2 }, *).

4.7 Let (R+, ⊗) is an algebraic structure where operation ⊗ is define as, a ⊗ b = ab/2 for all a, b ∈

R+, then (R+, ⊗) is an Abelian group.

4.8 Consider M AA

= AA 22

F

HG

I

KJ ×

Define a set of matrices, where A is nonzero real number. Then prove Algebraic structure (M,

*) where * is ordinary multiplication operation is a group.

4.9 Prove that nth root of unity forms a group under ordinary multiplication. Show also it is an

Abelian group.

4.10 Find all the subgroups of X generated by the permutations

1234

2314

F

HG

I

KJ and

abc d

3249

F

HG

I

KJ

(i)(ii)

4.11 Let (X, *) be a group and y ∈ X. Let f : X → X be given by f(x) = y * x * y–1 for all x ∈ X then show

that f is an isomorphism of X into X.

4.12 If an Abelian group has subgroups of orders m and n, then show that it has a subgroup whose

order is the least common multiple of m and n.

4.13 Define the left coset and the right coset. If (I, +) is a additive group of integers and the set

H = {...... – 6, – 3, 0, 3, 6, ......} then show that

I = (H + 0) ∪ (H + 1) ∪ (H + 2).

4.14 Define a ring with unity. Is the set of even integers i.e., J = 2I = {...... – 4, – 2, 0, 2, 4, ......} is a

ring with unity under the binary operation addition and multiplication. Is the ring (J, +, •) is

commutative also.

4.15 What do you understand by zero divisors of a ring? Give an example of ring with zero divisors

and without zero divisors.

4.16 Define a field. Prove that ({0, 1, 2, 3, 4}, + 5 , * 5 ) is a finite field.

4.17 What is a skew field? Is every skew field is a field. If not, give an example.