BMath Bachelor of Mathematics Abstract Algebra and Group Theory 2

Question 1

Which of the following is an example of a commutative ring that is NOT an integral domain?

Accepted Answer

Z_6

Answer

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?

Accepted Answer

Every nonzero element has a multiplicative inverse

Answer

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:

Accepted Answer

The set of all g ∈ G such that φ(g) = e_H

Answer

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:

Accepted Answer

G/ker(φ) ≅ Im(φ)

Answer

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?

Accepted Answer

A_n is a normal subgroup of S_n with index 2

Answer

A_n consists of all even permutations, is normal in S_n, and has index 2 because |A_n| = n!/2.

BMath Bachelor of Mathematics Practice Test