Ultimate Abstract Algebra & Group Theory Exam Pack:
Verified Multiple Choice Questions with Detailed Explanations
Cornell University
This premium document features highly tested, non-repetitive multiple-
choice questions spanning critical Abstract Algebra concepts, including
normal subgroups, homomorphisms, and Sylow theorems. Every single
question is paired with a bolded, verified answer and a bolded, in-depth
explanation to provide immediate clarity and reinforce complex
mathematical proofs. Save time and guarantee top grades with this
meticulously organized
Set Theory, Relations, and Functions
Q1: Let A,B,CA,B,C be sets such that A⊆BA⊆B. Which of the following is
always true?
A) A∪C⊆B∪CA∪C⊆B∪C
B) A∩C⊆B∩CA∩C⊆B∩C
C) Both A and B
D) Neither A nor B
Answer: C
Explanation: Since A⊆BA⊆B, every element of AA is in BB. Thus,
A∪C⊆B∪CA∪C⊆B∪C and A∩C⊆B∩CA∩C⊆B∩C both hold.
,Q2: For any sets A,B,CA,B,C, which of the following represents the
distributive law?
A) A∩(B∪C)=(A∩B)∪(A∩C)A∩(B∪C)=(A∩B)∪(A∩C)
B) A∪(B∩C)=(A∪B)∩(A∪C)A∪(B∩C)=(A∪B)∩(A∪C)
C) Both A and B
D) A∩(B∪C)=(A∪B)∩CA∩(B∪C)=(A∪B)∩C
Answer: C
Explanation: Both intersection distributes over union and union
distributes over intersection in set theory.
Q3: If f:A→Bf:A→B is a function and g:B→Ag:B→A such that
g(f(a))=ag(f(a))=a for all a∈Aa∈A, then which statement is true?
A) ff must be surjective
B) ff must be injective
C) gg must be surjective
D) gg must be bijective
Answer: B
Explanation: The condition g(f(a))=ag(f(a))=a implies ff has a left inverse,
which means ff is injective. It does not guarantee surjectivity.
Q4: Every function from a finite set onto itself must be:
A) One-to-one
B) Not one-to-one
C) Always bijective
D) Both A and C
Answer: D
Explanation: For a finite set, an onto function must also be one-to-one
(pigeonhole principle), hence bijective.
,Q5: The characteristic of the ring 2Z2Z is:
A) 0
B) 2
C) 4
D) Infinite
Answer: A
Explanation: A ring has characteristic nn if n⋅a=0n⋅a=0 for all aa. For
2Z2Z, k⋅2=0k⋅2=0 implies k=0k=0, so the characteristic is 0.
Q6: Which of the following is a binary operation on Z∗Z∗?
A) a∗b=a−ba∗b=a−b
B) a∗b=aba∗b=ab
C) a∗b=max(a,b)a∗b=max(a,b)
D) a∗b=a/ba∗b=a/b
Answer: C
Explanation: max(a,b)max(a,b) is always an integer, so it is closed on
Z∗Z∗. Subtraction and division may yield non-integers.
Q7: In the group (Z,+)(Z,+), which of the following is a subgroup?
A) Odd integers
B) Even integers
C) Positive integers
D) Prime numbers
Answer: B
Explanation: Even integers are closed under addition, contain 0, and have
inverses, satisfying subgroup criteria.
, Q8: Which of the following structures is NOT a group?
A) Integers under addition
B) Non-zero real numbers under multiplication
C) Natural numbers under addition
D) Symmetries of a square under composition
Answer: C
Explanation: Natural numbers lack additive inverses, so they do not form
a group.
Q9: The identity element in a group is:
A) Unique and may not exist
B) Always 1
C) Always 0
D) Unique and always exists
Answer: D
Explanation: By group axioms, every group has a unique identity element.
Q10: A group is abelian if:
A) It has no identity
B) Every element has an inverse
C) The operation is commutative
D) It has finite elements
Answer: C
Explanation: Abelian groups satisfy a∗b=b∗aa∗b=b∗a for all elements.
Groups, Subgroups, and Cyclic Groups
Verified Multiple Choice Questions with Detailed Explanations
Cornell University
This premium document features highly tested, non-repetitive multiple-
choice questions spanning critical Abstract Algebra concepts, including
normal subgroups, homomorphisms, and Sylow theorems. Every single
question is paired with a bolded, verified answer and a bolded, in-depth
explanation to provide immediate clarity and reinforce complex
mathematical proofs. Save time and guarantee top grades with this
meticulously organized
Set Theory, Relations, and Functions
Q1: Let A,B,CA,B,C be sets such that A⊆BA⊆B. Which of the following is
always true?
A) A∪C⊆B∪CA∪C⊆B∪C
B) A∩C⊆B∩CA∩C⊆B∩C
C) Both A and B
D) Neither A nor B
Answer: C
Explanation: Since A⊆BA⊆B, every element of AA is in BB. Thus,
A∪C⊆B∪CA∪C⊆B∪C and A∩C⊆B∩CA∩C⊆B∩C both hold.
,Q2: For any sets A,B,CA,B,C, which of the following represents the
distributive law?
A) A∩(B∪C)=(A∩B)∪(A∩C)A∩(B∪C)=(A∩B)∪(A∩C)
B) A∪(B∩C)=(A∪B)∩(A∪C)A∪(B∩C)=(A∪B)∩(A∪C)
C) Both A and B
D) A∩(B∪C)=(A∪B)∩CA∩(B∪C)=(A∪B)∩C
Answer: C
Explanation: Both intersection distributes over union and union
distributes over intersection in set theory.
Q3: If f:A→Bf:A→B is a function and g:B→Ag:B→A such that
g(f(a))=ag(f(a))=a for all a∈Aa∈A, then which statement is true?
A) ff must be surjective
B) ff must be injective
C) gg must be surjective
D) gg must be bijective
Answer: B
Explanation: The condition g(f(a))=ag(f(a))=a implies ff has a left inverse,
which means ff is injective. It does not guarantee surjectivity.
Q4: Every function from a finite set onto itself must be:
A) One-to-one
B) Not one-to-one
C) Always bijective
D) Both A and C
Answer: D
Explanation: For a finite set, an onto function must also be one-to-one
(pigeonhole principle), hence bijective.
,Q5: The characteristic of the ring 2Z2Z is:
A) 0
B) 2
C) 4
D) Infinite
Answer: A
Explanation: A ring has characteristic nn if n⋅a=0n⋅a=0 for all aa. For
2Z2Z, k⋅2=0k⋅2=0 implies k=0k=0, so the characteristic is 0.
Q6: Which of the following is a binary operation on Z∗Z∗?
A) a∗b=a−ba∗b=a−b
B) a∗b=aba∗b=ab
C) a∗b=max(a,b)a∗b=max(a,b)
D) a∗b=a/ba∗b=a/b
Answer: C
Explanation: max(a,b)max(a,b) is always an integer, so it is closed on
Z∗Z∗. Subtraction and division may yield non-integers.
Q7: In the group (Z,+)(Z,+), which of the following is a subgroup?
A) Odd integers
B) Even integers
C) Positive integers
D) Prime numbers
Answer: B
Explanation: Even integers are closed under addition, contain 0, and have
inverses, satisfying subgroup criteria.
, Q8: Which of the following structures is NOT a group?
A) Integers under addition
B) Non-zero real numbers under multiplication
C) Natural numbers under addition
D) Symmetries of a square under composition
Answer: C
Explanation: Natural numbers lack additive inverses, so they do not form
a group.
Q9: The identity element in a group is:
A) Unique and may not exist
B) Always 1
C) Always 0
D) Unique and always exists
Answer: D
Explanation: By group axioms, every group has a unique identity element.
Q10: A group is abelian if:
A) It has no identity
B) Every element has an inverse
C) The operation is commutative
D) It has finite elements
Answer: C
Explanation: Abelian groups satisfy a∗b=b∗aa∗b=b∗a for all elements.
Groups, Subgroups, and Cyclic Groups
