This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Permutation Groups”.

1. Consider an integer 23 such that 23 >= 3p for a 2p-cycle in a permutation group, then p is ___________

a) odd prime

b) even prime

c) rational number

d) negative prime

View Answer

Explanation: Let n an integer such that n>=3p and m is a 2p-cycle in the permutation group, then p is an odd prime.

2. Suppose K_{m}={P∈S_{m}|, |P| is odd prime}. Determine the set for which m≥3 Km a subgroup of S_{m}.

a) {3, 5, 7, 11, 13, …}

b) {-14, -8, -3, 0, 3, 8, 14}

c) {2, 4, 6, 8, 10, 12}

d) {12, 25, 56, 78, 134,…}

View Answer

Explanation: Since K

_{m}is a subset of S

_{m}, then the set will be {3, 5, 7, 11, 13, …}.

3. The dihedral group having order 6 can have degree _____________

a) 3

b) 26

c) 326

d) 208

View Answer

Explanation: A symmetric group on a set of three elements is said to be the group of all permutations of a three-element set. It is a dihedral group of order six having degree three.

4. Let (z, *) is a group with x*y=x+y-2 then inverse of x is ___________

a) -(x+4)

b) (x^{2}+6)

c) (x+y)/5

d) (3y+4x^{2})

View Answer

Explanation: Let, Identity element I, x*I = I*x = x ⇒ x = x + I – 2 ⇒ I = 2. Inverse of x is x

^{-1}

Now, x*x

^{-1}= I

⇒ x + x

^{-1}– 2 = 2

⇒ x

^{-1}= -(x+4).

5. Let X be a n-square matrix such that Y = X + 8I. Which of the following property will exist?

a) idempotent

b) Y transpose is nilpotent

c) X nilpotent

d) Y inverse

View Answer

Explanation: Suppose, we have a matrix

\(a=\begin{bmatrix}

1 & 0\\

2 & 1\\

\end{bmatrix} \) then Y

^{2}will not resulting in Y, hence it is not idempotent, Y

^{2}is not 0 and so it is not nilpotent. But, as Y is a square matrix, by the property inverse will exist in this case.

6. Suppose, M is a lower triangular matrix with all diagonal entries zero. The resultant matrix of M+I will be ___________

a) idempotent

b) singular

c) nilpotent

d) inverse

View Answer

Explanation: Since, M is a lower triangular matrix with diagonal elements zero, then we add I and it will result in a lower triangular matrix with all diagonal entries 1. Thus, all eigenvalues M+I are non zero (eigenvalues of triangular matrix is the diagonal elements). So, determinant will never be zero. Hence, the matrix can have inverse property.

7. If Y^{98} (a raised to the power of 5) = 0 and Y is a 97-square matrix. Determine the value of Y^{97}.

a) I+Y

b) -Y+3

c) 0

d) Y^{2}

View Answer

Explanation: Question does not provide any notion of existing an inverse property or related to rank matrix. Hence, by considering zero matrix as Y and that satisfy all the constraints.

8. If 54^{th} row of a 67^{th} row matrix is linearly independent with each other then find the rank of the matrix.

a) 61

b) 54

c) 187

d) 32

View Answer

Explanation: If k

^{th}row of a matrix with n

^{th}row is linearly independent then the rank of that matrix is k. If we take the transpose of a matrix the rank does not change. Hence, the answer is 54 in this case.

9. Let M be an 4×4 matrix with real entries such that M^{k}=0, for some k≥1. Find the determinant value of (I+M), where, I be the 4 x 4 identity matrix.

a) 72

b) 1

c) 4

d) 36

View Answer

Explanation: By cayley hamilton theorem, M

^{4}= 0. So, characteristic equation should be λ*4=0 and after solving we get 0 for every eigen value. Eigen values of (I+M) = Individual Eigen value of 1+m. So all the eigen values of (I+M) are 1 and Det(I+A) = 1.

10. Suppose (2, 5, 8, 4) and (3, 6) are the two permutation groups that form cycles. What type of permutation is this?

a) odd

b) even

c) acyclic

d) prime

View Answer

Explanation: There are four permutations (2, 5), (2, 8), (2, 4) and (3, 6) and so it is an even permutation.

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