Abstract Algebra MCQs

By: Prof. Dr. Fazal Rehman | Last updated: January 31, 2025

Q1: Which of the following is a group under addition?

(A) ℕ
(B) ℤ
(C) ℝ⁺
(D) ℚ⁺

Answer: (B) ℤ

Q2: Which of the following is a field?

(A) ℤ
(B) ℝ
(C) ℕ
(D) ℚ

Answer: (D) ℚ

Q3: Which of the following is an abelian group?

(A) The set of integers under multiplication
(B) The set of non-zero real numbers under multiplication
(C) The set of 2×2 invertible matrices under matrix multiplication
(D) The set of integers under addition

Answer: (D) The set of integers under addition

Q4: If a group has an identity element e, which of the following is true for any element a in the group?

(A) a * e = a
(B) a * e = e
(C) a * e = e * a
(D) a * a = e

Answer: (A) a * e = a

Q5: Which of the following sets is not a subgroup of ℝ under addition?

(A) ℤ
(B) ℝ
(C) ℝ⁺
(D) ℚ

Answer: (C) ℝ⁺

Q6: What is the order of the group ℤ₆ under addition modulo 6?

(A) 6
(B) 3
(C) 2
(D) 1

Answer: (A) 6

Q7: A ring is a set R with two binary operations. Which of the following is a necessary condition for R to be a ring?

(A) R must have an identity element for multiplication.
(B) R must be abelian under addition.
(C) R must be commutative under multiplication.
(D) R must have an identity element for addition.

Answer: (B) R must be abelian under addition.

Q8: Which of the following is an example of a non-abelian group?

(A) ℤ
(B) ℝ⁺
(C) The set of 2×2 invertible matrices under matrix multiplication
(D) The set of real numbers under addition

Answer: (C) The set of 2×2 invertible matrices under matrix multiplication

Q9: If a ring has a multiplicative identity, then it is called:

(A) A field
(B) A division ring
(C) A commutative ring
(D) A unital ring

Answer: (D) A unital ring

Q10: Which of the following is true about a cyclic group?

(A) It contains only one element
(B) It is isomorphic to ℤ
(C) It is always finite
(D) All of the above

Answer: (B) It is isomorphic to ℤ

More MCQs on Algebra (Collected from Past Papers)

  1. Elementary Algebra – MCQs
  2. Algebraic Expressions – MCQs
  3. Algebraic Identities – MCQs
  4. Commutative Algebra – MCQs
  5. Linear Equations – MCQs
  6. Graphing Linear Equations – MCQs
  7. Inequalities in Algebra – MCQs
  8. Absolute Value Inequalities – MCQs
  9. Exponent – MCQs
  10. Exponential Functions – MCQs
  11. Logarithms – MCQs
  12. Polynomials – MCQs
  13. Factoring Methods – MCQs
  14. Polynomial Arithmetic – MCQs
  15. Quadratic Equation – MCQs
  16. Linear Algebra – MCQs
  17. Matrices in Algebra – MCQs
  18. Functions in Algebra – MCQs
  19. Sequences in Algebra – MCQs
  20. Arithmetic in Algebra – MCQs
  21. Combining Like Terms in Algebra – MCQs
  22. Abstract Algebra – MCQs
  23. Sets in Algebra – MCQs
  24. Algebra Calculator – MCQs
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