BMath Bachelor of Mathematics Abstract Algebra and Group Theory 2 — Questions and Answers
Question 1: Which of the following is an example of a commutative ring that is NOT an integral domain?
- Z (integers)
- Z_5
- Z_6 (Correct answer)
- Q (rationals)
Correct answer: Z_6
Z_6 has zero divisors since 2 × 3 = 6 ≡ 0 (mod 6) with 2 ≠ 0 and 3 ≠ 0, violating the integral domain condition.
Question 2: A field is a commutative ring in which which additional property holds?
- Every element has an additive inverse
- Every nonzero element has a multiplicative inverse (Correct answer)
- The characteristic is always zero
- Multiplication need not be associative
Correct answer: Every nonzero element has a multiplicative inverse
A field is a commutative ring where every nonzero element has a multiplicative inverse, enabling division by any nonzero element.
Question 3: The kernel of a group homomorphism φ: G → H is defined as:
- The image Im(φ) inside H
- The set of all g ∈ G such that φ(g) = e_H (Correct answer)
- A subgroup of H
- The center of G
Correct answer: The set of all g ∈ G such that φ(g) = e_H
ker(φ) = {g ∈ G : φ(g) = e_H}, the preimage of the identity element of H under φ.
Question 4: The First Isomorphism Theorem states that if φ: G → H is a group homomorphism, then:
- G ≅ H
- G/ker(φ) ≅ Im(φ) (Correct answer)
- G ≅ ker(φ) × Im(φ)
- ker(φ) ≅ H
Correct answer: G/ker(φ) ≅ Im(φ)
The First Isomorphism Theorem establishes that the quotient group G/ker(φ) is isomorphic to the image Im(φ) of the homomorphism.
Question 5: Which statement about the alternating group A_n is TRUE?
- A_n = S_n for all n ≥ 2
- A_n consists of all odd permutations in S_n
- A_n is a normal subgroup of S_n with index 2 (Correct answer)
- |A_n| = n!
Correct answer: A_n is a normal subgroup of S_n with index 2
A_n consists of all even permutations, is normal in S_n, and has index 2 because |A_n| = n!/2.
Question 6: The polynomial ring F[x] over a field F is best described as:
- Always a field itself
- A principal ideal domain (PID) (Correct answer)
- Never an integral domain
- A group under polynomial multiplication
Correct answer: A principal ideal domain (PID)
F[x] over a field F is a principal ideal domain because every ideal in F[x] is generated by a single polynomial (the GCD of the ideal's elements).
Question 7: The center Z(G) of a group G is defined as:
- The group G itself
- The set of elements that commute with every element of G (Correct answer)
- A normal subgroup only when G is abelian
- The trivial subgroup {e}
Correct answer: The set of elements that commute with every element of G
Z(G) = {z ∈ G : zg = gz for all g ∈ G} is the set of all elements that commute with every element of the group.
BMath Bachelor of Mathematics Abstract Algebra and Group Theory 2
Which of the following is an example of a commutative ring that is NOT an integral domain?