Q.1 Which of the following is a group under addition?
Natural numbers
Integers
Positive integers
Even natural numbers
Explanation - The set of integers under addition satisfies closure, associativity, identity (0), and inverse (-a for any a).
Correct answer is: Integers
Q.2 The identity element in a group G is:
Always unique
Not unique
May not exist
Always multiple
Explanation - Every group has a unique identity element by definition of group axioms.
Correct answer is: Always unique
Q.3 Which of the following is NOT an abelian group?
Integers under addition
Real numbers under addition
Nonzero real numbers under multiplication
2x2 invertible matrices under multiplication
Explanation - Matrix multiplication is not commutative in general, so GL(2, R) is not abelian.
Correct answer is: 2x2 invertible matrices under multiplication
Q.4 Which property is NOT required for a group?
Closure
Commutativity
Associativity
Existence of inverse
Explanation - Commutativity is required only for abelian groups, not for all groups.
Correct answer is: Commutativity
Q.5 The set of even integers under addition forms:
A group
Not a group
A semigroup but not a group
A monoid but not a group
Explanation - Even integers under addition satisfy all group axioms with identity 0 and inverses present.
Correct answer is: A group
Q.6 Which of the following is a finite group?
Integers under addition
Real numbers under multiplication
Integers modulo n under addition
Rational numbers under addition
Explanation - The group Z_n under addition mod n is finite with n elements.
Correct answer is: Integers modulo n under addition
Q.7 Which group is cyclic?
Integers under addition
Real numbers under addition
2x2 invertible matrices
Rational numbers under addition
Explanation - The group (Z, +) is cyclic, generated by 1 or -1.
Correct answer is: Integers under addition
Q.8 Order of an element in a group refers to:
The number of elements in the group
The number of operations possible
The smallest positive integer n such that a^n = e
The position of the element
Explanation - The order of an element is defined as the least positive integer for which repeated operation gives identity.
Correct answer is: The smallest positive integer n such that a^n = e
Q.9 Which of these sets forms a monoid but not a group under multiplication?
Integers
Nonzero rational numbers
Nonzero real numbers
Natural numbers
Explanation - Natural numbers under multiplication have closure, associativity, and identity (1), but lack inverses.
Correct answer is: Natural numbers
Q.10 If a group has only one element, it is called:
Trivial group
Abelian group
Finite group
Infinite group
Explanation - The group with only the identity element is called the trivial group.
Correct answer is: Trivial group
Q.11 Which of the following is true for every subgroup H of G?
H must be abelian
H must contain the identity of G
H must be infinite
H must equal G
Explanation - Every subgroup contains the identity of the parent group.
Correct answer is: H must contain the identity of G
Q.12 The set of invertible 2x2 matrices over real numbers forms a:
Group under addition
Group under multiplication
Semigroup under addition
Cyclic group
Explanation - GL(2,R) is a group under multiplication of matrices since invertible matrices have inverses.
Correct answer is: Group under multiplication
Q.13 Which of these is NOT a subgroup of (Z, +)?
2Z
3Z
Even integers
Odd integers
Explanation - Odd integers do not contain 0 (identity) and are not closed under addition.
Correct answer is: Odd integers
Q.14 Which of the following is true for every cyclic group?
It is always finite
It is always infinite
It can be finite or infinite
It is not a group
Explanation - Cyclic groups may be finite (Z_n) or infinite (Z).
Correct answer is: It can be finite or infinite
Q.15 Which algebraic structure has only closure and associativity?
Monoid
Group
Semigroup
Ring
Explanation - A semigroup requires only closure and associativity.
Correct answer is: Semigroup
Q.16 The Klein four-group has how many elements?
2
3
4
5
Explanation - The Klein four-group V_4 has four elements {e, a, b, c}.
Correct answer is: 4
Q.17 Which structure has closure, associativity, identity but not inverses?
Semigroup
Monoid
Group
Field
Explanation - A monoid has closure, associativity, and identity but inverses are not required.
Correct answer is: Monoid
Q.18 What is the order of the group (Z_6, +)?
5
6
7
Infinite
Explanation - The group Z_6 under addition modulo 6 has 6 elements.
Correct answer is: 6
Q.19 Which of the following is a non-abelian finite group?
Z_5
Z_2 × Z_2
S_3
Integers under addition
Explanation - The symmetric group S_3 is non-abelian.
Correct answer is: S_3
Q.20 A group homomorphism preserves:
Only inverses
Only identity
Only closure
Group operation
Explanation - A homomorphism f satisfies f(ab) = f(a)f(b).
Correct answer is: Group operation
Q.21 Kernel of a group homomorphism is always a:
Subgroup
Coset
Ring
Field
Explanation - The kernel of a homomorphism is a subgroup of the domain.
Correct answer is: Subgroup
Q.22 Which of the following is true about every subgroup?
It must be finite
It must contain the identity
It must be abelian
It must equal the group
Explanation - Identity is always present in a subgroup.
Correct answer is: It must contain the identity
Q.23 The order of any subgroup divides:
The order of the group
The number of generators
The order of another subgroup
Infinity
Explanation - By Lagrange’s theorem, order of subgroup divides the order of the group.
Correct answer is: The order of the group
Q.24 Which is the smallest non-cyclic group?
Z_2
Z_3
V_4
Z_4
Explanation - The Klein four-group V_4 is the smallest non-cyclic group.
Correct answer is: V_4
Q.25 Direct product of two cyclic groups is always:
Cyclic
Non-cyclic
Sometimes cyclic
Never cyclic
Explanation - Direct product of cyclic groups is cyclic if and only if their orders are coprime.
Correct answer is: Sometimes cyclic