PG TRB -Algebra FREE Test / Quiz

Algebra Quiz - 30 MCQs

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1. Which of the following is a cyclic group?
   (ℤ, +)
   (ℚ, +)
   (ℝ, +)
   (ℚ*, ×)
   


2. By Cayley’s theorem, every finite group is isomorphic to a subgroup of:
   A cyclic group
   A permutation group
   A dihedral group
   An abelian group
   


3. Let G be a finite group and p divides |G|. Sylow's theorems guarantee existence of:
   Normal p-subgroup
   Cyclic p-subgroup
   Subgroup of order p^k for some maximal k
   Simple p-subgroup
   


4. Every subgroup of a cyclic group is:
   Cyclic
   Abelian
   Simple
   Normal
   


5. The alternating group A₅ is:
   Simple
   Cyclic
   Abelian
   Finite abelian
   


6. If H is a normal subgroup of G, then for all g in G:
   gHg^-1 = H
   gH = Hg
   g^-1Hg = H
   None of these
   


7. Which of the following is NOT an integral domain?
   ℤ₆
   ℤ
   ℤ₅
   ℚ[x]
   


8. The field of quotients of ℤ[x] is:
   ℤ
   ℚ
   ℚ(x)
   ℝ
   


9. Every Euclidean domain is a:
   Field
   Unique factorization domain (UFD)
   Division ring
   Principal ideal domain (PID)
   


10. In a finite-dimensional vector space, any two bases have:
    Equal number of vectors
    Different number of vectors
    Some dependent vectors
    Same vectors
    


11. Which group is NOT abelian?
    (ℤ, +)
    (ℝ, +)
    S₃
    (ℚ, +)
    


12. The only ideals of a field are:
    {0} and the field itself
    Prime ideals
    Maximal ideals
    {0} only
    


13. Which of the following is a simple group?
    S₃
    ℤ₅
    A₅
    ℤ
    


14. Every field is a:
    Euclidean domain
    UFD
    PID
    All of these
    


15. A group of order 21 has a normal subgroup of order:
    3
    7
    21
    Both A and B
    


16. If p is a prime, every finite field of characteristic p has order:
    p
    pⁿ for some n ≥ 1
    2p
    Any integer
    


17. Which statement is true about vector spaces?
    Every subspace has a basis
    Every basis is unique
    Every linearly independent set is a basis
    Every spanning set is a basis
    


18. A linear operator T on a vector space V is nilpotent if:
    Tⁿ = 0 for some n
    T is invertible
    T has only zero eigenvalues
    Tⁿ = I for all n
    


19. A Hermitian matrix always has:
    Complex eigenvalues
    Real eigenvalues
    Purely imaginary eigenvalues
    Zero trace
    


20. The matrix A = [[0, 1], [0, 0]] is:
    Diagonalizable
    Nilpotent
    Hermitian
    Unitary
    


21. The map ϕ: ℤ → ℤ/5ℤ given by ϕ(n) = n mod 5 is a:
    Homomorphism
    Isomorphism
    Automorphism
    None
    


22. In a field, every nonzero element has:
    A unique inverse
    An additive inverse only
    No inverse
    Multiple inverses
    


23. For a matrix A, if there exists an invertible matrix P such that P^-1AP = D where D is diagonal, then A is:
    Nilpotent
    Diagonalizable
    Orthogonal
    None
    


24. Which statement is true about inner product spaces?
    Every orthonormal set is a basis
    All bases are orthonormal
    All orthogonal sets are bases
    None
    


25. The Jordan canonical form of a nilpotent matrix consists of:
    All diagonal zero blocks
    Diagonal 1's
    Upper-triangular blocks with zero diagonal
    No zero blocks
    


26. If a matrix is both unitary and Hermitian, then it is:
    The identity
    Diagonalizable with eigenvalues ±1 only
    Singular
    None
    


27. The rank of a matrix equals:
    The number of independent rows
    The number of columns
    Order of the determinant
    Trace
    


28. A vector space and its dual space always have:
    Equal dimensions
    Different dimensions
    Dual is never a vector space
    Dual has smaller dimension
    


29. Let f(x) be an irreducible polynomial in ℚ[x], then the ideal (f(x)) in ℚ[x] is:
    Maximal
    Prime
    Both A and B
    Neither
    


30. An element α in field extension K/F is algebraic over F if:
    It is a root of a nonzero polynomial with coefficients in F
    For every nonzero polynomial in K[x], α is a root
    α has no minimal polynomial
    None
    

Answer Key:

1. Question 1: A
2. Question 2: B
3. Question 3: C
4. Question 4: A
5. Question 5: A
6. Question 6: A
7. Question 7: A
8. Question 8: C
9. Question 9: D
10. Question 10: A
11. Question 11: C
12. Question 12: A
13. Question 13: C
14. Question 14: D
15. Question 15: D
16. Question 16: B
17. Question 17: A
18. Question 18: A
19. Question 19: B
20. Question 20: B
21. Question 21: A
22. Question 22: A
23. Question 23: B
24. Question 24: A
25. Question 25: C
26. Question 26: B
27. Question 27: A
28. Question 28: A
29. Question 29: C
30. Question 30: A