Discrete Mathematics Multiple Choice Questions 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
Answer: a
Clarification: 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 Km={P∈Sm|, |P| is odd prime}. Determine the set for which m≥3 Km a subgroup of Sm.
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,…}
Answer: a
Clarification: Since Km is a subset of Sm, 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
Answer: a
Clarification: 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) (x2+6)
c) (x+y)/5
d) (3y+4x2)
Answer: a
Clarification: 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
Answer: b
Clarification: Suppose, we have a matrix
(a=begin{bmatrix}
1 & 0\
2 & 1\
end{bmatrix} )
then Y2 will not resulting in Y, hence it is not idempotent, Y2 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
Answer: b
Clarification: 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 Y98 (a raised to the power of 5) = 0 and Y is a 97-square matrix. Determine the value of Y97.
a) I+Y
b) -Y+3
c) 0
d) Y2
Answer: c
Clarification: 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 54th row of a 67th row matrix is linearly independent with each other then find the rank of the matrix.
a) 61
b) 54
c) 187
d) 32
Answer: b
Clarification: If kth row of a matrix with nth 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 Mk=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
Answer: b
Clarification: By cayley hamilton theorem, M4 = 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
Answer: b
Clarification: There are four permutations (2, 5), (2, 8), (2, 4) and (3, 6) and so it is an even permutation.