Explore topic-wise InterviewSolutions in .

Right choice is (C) 1

Explanation: The group of real matrices with determinant 1 is a subgroup of the group of invertible real matrices, both equipped with matrix MULTIPLICATION. It has to be SHOWN that the product of two matrices with determinant 1 is ANOTHER matrix with determinant 1, but this is immediate from the multiplicative property of the determinant. This group is usually denoted by(n, R).

Correct answer is (d) 4

The BEST I can explain: For any FINITE group G, the order (number of ELEMENTS) of every subgroup L of G divides the order of G. G has 8 elements. Factors of 8 are 1, 2, 4 and 8. Since given the size of L is at LEAST 3(1 and 2 eliminated) and not equal to G(8 eliminated), the only size left is 4. Size of L is 4.

The CORRECT answer is (c) a trivial group

The best EXPLANATION: The SUBGROUPS of any given group FORM a complete lattice under inclusion termed as a lattice of subgroups. If o is the Identity element of a group(G), then the trivial group(o) is the MINIMUM subgroup of that group and G is the maximum subgroup.

The correct option is (a) Identity ELEMENT

To explain: Let G be a GROUP under a binary operation * and a SUBSET H of G is called a SUBGROUP of G if H forms a group under the operation *. The trivial subgroup of any group is the subgroup consisting of only the Identity element.

The correct option is (c) a^2=a*a=a

The best I can explain: An algebraic structure with a SINGLE ASSOCIATIVE binary OPERATION and an Identity element are termed as a monoid. It is studied in semigroup THEORY. An element x in a monoid is called IDEMPOTENT if a^2 = a*a = a.

Correct choice is (a) lie GROUPS

Explanation: A symmetry GROUP can encode symmetry FEATURES of a geometrical object. The group consists of the set of TRANSFORMATIONS that leave the object unchanged. Lie groups are such symmetry groups used in the standard model of particle physics.

Right ANSWER is (b) MONOID

Easy explanation: LET P(S) is a commutative semigroup has the identity e, since e*A=A=A*e for any element A BELONGS to P(S). Hence, P(S) is a monoid.

The correct answer is (d) singular

Easy EXPLANATION: By the DEFINITION, an n×n matrix A is said to be singular if there exists a nonzero vector v such that Av=0. OTHERWISE, it is known that A is a nonsingular matrix.

The CORRECT option is (b) singular

For explanation I WOULD say: The rational numbers are an extension of the integer numbers in which each non-zero number has an inverse under multiplication. A 3 × 3 MATRIX may or may not have an inverse under matrix multiplication. The matrices which do not have MULTIPLICATIVE inverses are termed as singular matrices.

Correct OPTION is (b) IDENTITY matrix

To explain: Since X is idempotent, we have X^2=X. As X is NONSINGULAR, it is invertible. THUS, the INVERSE matrix X^-1 exists. Then we have, I=X^-1X = X^-1X2=IX=X.

Right choice is (c) inverse

To elaborate: By the definition of all elements of a GROUP have an inverse. For an ELEMENT, a in a group G, an inverse of a is an element b such that ab=e, where e is the identity in the group. The inverse of an element is unique and USUALLY denoted as -a.

Right CHOICE is (C) 1

For explanation I would say: 1 is the multiplicative identity of natural NUMBERS as a⋅1=a=1⋅a ∀a∈N. Thus, 1 is the identity of multiplication for the set of integers(Z), set of rational numbers(Q), and set of real numbers(R).

Correct option is (a) closure PROPERTY

To EXPLAIN I would say: The SET of even natural numbers is closed by the addition as the sum of any TWO of them produces another even number. Hence, this closed set satisfies the closure property.

The correct answer is (c) 72

The explanation: We need to find the NUMBER of co-primes of 73 which are LESS than 73. As 73 itself is a prime, all the numbers less than that are co-prime to it and it makes a group of order 72 then it can be of {1, 3, 5, 7, 11….}.

Right choice is (b) closure, ASSOCIATIVITY, inverse and identity

For explanation: Closure for all a, b∈G, the result of the operation, a+b, is also in G. SINCE there is one ELEMENT, hence a=b=0, and a+b=0+0=0∈G. Hence, closure property is satisfied. Associative for all a, b, c∈G, (a+b)+c=a+(b+c). For example, a=b=c=0. Hence (a+b)+c=a+(b+c)

⟹(0+0)+0=0+(0+0)⟹0=0. Hence, associativity property is satisfied. Suppose for an element e∈G such that, there EXISTS an element a∈G and so the equation e+a=a+e=a holds. Such an element is unique, the identity element property is satisfied. For example, a=e=0. Hence e+a = a+e⟹0+0=0+0⟹0=a. Hence e=0 is the identity element. For each a∈G, there exists an element b∈G (denoted as a-1), such that a+b=b+a=e, where e is the identity element. The inverse element is 0 as the addition of 0 with 0 will be 0, which is also an identity element of the structure.

Right option is (a) 1

For explanation: We KNOW that GCD(56, 123)=1. So, the VALUE of |M⋂N|=1.

The correct choice is (b) semigroups

Easy EXPLANATION: Here, B1 is the group and IDENTITY ELEMENT is 0, MEANS for all a∈B1, a+n.0=a. As a

The CORRECT option is (a) ABELIAN GROUP

Easiest explanation: SINCE * closed operation, a*b belongs to X. Hence, it is an abelian group.

Correct ANSWER is (b) associative

The explanation: For any three ELEMENTS(numbers) a, b and c associative property DESCRIBES a × ( b × c ) = ( a × b ) × c[for MULTIPLICATION]. Hence associative property is TRUE for multiplication and it is true for multiplication also.

The correct answer is (b) 5

To elaborate: Given, (A7, ⊗7)=({1, 2, 3, 4, 5, 6}, ⊗7) and the union of two sub groups X and Y, X={1, 3, 6} Y={2, 3, 5} is X∪Y={1, 2, 3, 5, 6} i.e., 5. Here, the ORDER of the union can not be divided by order of the GROUP.

Correct answer is (b) Homomorphism exists

The best EXPLANATION: Let T be the group of real numbers under ADDITION, and let T’ be the group of positive real numbers under multiplication. The mapping F: T -> T’ defined by f(a)=2*a is a homomorphism because f(a+b)=2a+b = 2a*2b = f(a)*f(b). Again f is also one-to-one and onto T and T’ are isomorphic.

Right choice is (c) COMMUTATIVE semigroup

The explanation: LET x and y belongs to a GROUP G.Here closure and associativity axiom holds simultaneously. Let e be an element in G such that x * e = x then x+e+xe=a => e(1+x)=0 => e = 0/(1+x) = 0. So, identity axiom does not exist but commutative PROPERTY holds. THUS, (G,*) is a commutative semigroup.

The correct CHOICE is (b) f(a, b) = a/b

For explanation I would say: The FUNCTION f is a HOMOMORPHISM since f(x∗y)= f(AC, BD)= (ac)/(bd) = (a/b)(c/d) = f(x)f(y).

The correct option is (d) F(x * y) = f(x) * f(y)

For explanation I would say: Consider two semigroups (S,∗) and (S’,∗’). A FUNCTION f: S -> S’ is CALLED a semigroup homomorphism if f(a∗b) = f(a)∗f(b). Suppose f is also one-to-one and onto. Then f is called an isomorphism between S and S’ and S and S’ are SAID to be isomorphic semigroups.

Right choice is (b) a subgroup of F

The explanation is: Let F be the free semigroup on the set S = {m,n}. Let, E consist of all even words, i.e, words with even LENGTH and the CONCATENATION of two such words is ALSO even. Thus E is a SUBSEMIGROUP of F.

Correct OPTION is (C) Multiplication

Explanation: The set of nonzero rational numbers form an abelian group under multiplication. The NUMBER 1 is the identity element and q/p is the MULTIPLICATIVE inverse of the rational number p/q.

Right CHOICE is (B) a subsemigroup of (M, ×)

To explain: LET C and D be the set of EVEN and odd positive integers. Then, (C, ×) and (D, ×) are subsemigroups of (M, ×) since A and B are closed under MULTIPLICATION. On the other hand, (A, +) is a subsemigroup of (N, +) since A is closed under addition, but (B, +) is not a subsemigroup of (N, +) since B is not closed under addition.

Right option is (a) * is associative

Explanation: ‘∗’ can be defined by the formula a∗b = a for any a and b in S. HENCE, (a ∗ b)∗c = a∗c = a and a ∗(b ∗ c)= a ∗ b = a. THEREFORE, ”∗” is associative. Hence (S, ∗) is a semigroup. On the CONTRARY, * 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.

The correct choice is (C) Group axioms

To ELABORATE: The group axioms are ALSO called the group postulates. A group with an identity (that is, a monoid) in which every ELEMENT has an INVERSE is termed as semi group.

Right ANSWER is (d) Closure, associative, Identity, Inverse

To elaborate: A subgroup S is a subset of a group G (denoted by S <= G) if it holds the FOUR properties simultaneously – Closure, Associative, Identity and Inverse ELEMENT.

The CORRECT answer is (b) (x*y)=(y*x)

To elaborate: A group (M,*) is SAID to be ABELIAN if (x*y) = (x*y) for all x, y belongs to M. Thus Commutative PROPERTY should hold in a group.

Correct option is (c) 5

Easiest explanation: A group HOLDS FIVE properties SIMULTANEOUSLY –

i) Closure

ii) associative

iii) Commutative

iv) IDENTITY element

v) Inverse element.

The correct option is (a) ABELIAN group

Easy EXPLANATION: A cyclic group is ALWAYS an abelian group but every abelian group is not a cyclic group. For INSTANCE, the rational numbers under addition is an abelian group but is not a cyclic one.

Correct option is (d) (a*e)=(e*a)=a

To explain: A SEMIGROUP (S,*) is DEFINED as a monoid if there exists an ELEMENT e in S such that (a*e) = (e*a) = a for all a in S. This element is CALLED identity element of S w.r.t *.