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 (integers)

Answer

Z_5

Answer

Q (rationals)

Question 2

A field is a commutative ring in which which additional property holds?

Accepted Answer

Every nonzero element has a multiplicative inverse

Answer

Every element has an additive inverse

Answer

The characteristic is always zero

Answer

Multiplication need not be associative

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

The image Im(φ) inside H

Answer

A subgroup of H

Answer

The center of G

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

G ≅ H

Answer

G ≅ ker(φ) × Im(φ)

Answer

ker(φ) ≅ H

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 = S_n for all n ≥ 2

Answer

A_n consists of all odd permutations in S_n

Answer

|A_n| = n!

Question 6

The polynomial ring F[x] over a field F is best described as:

Accepted Answer

A principal ideal domain (PID)

Answer

Always a field itself

Answer

Never an integral domain

Answer

A group under polynomial multiplication

Question 7

The center Z(G) of a group G is defined as:

Accepted Answer

The set of elements that commute with every element of G

Answer

The group G itself

Answer

A normal subgroup only when G is abelian

Answer

The trivial subgroup {e}