Group Theory Quizzes: Master Group Structures and Properties

Question 1

Which of the following options is NOT a characteristic property of permutations?

Accepted Answer

All permutations are necessarily even.

Answer

Permutations can be combined through composition.

Answer

Permutations possess inverses.

Answer

Permutations can be represented using matrices.

Question 2

Which historical figure is recognized for their significant contributions to the development of group theory?

Accepted Answer

Évariste Galois

Answer

Niels Henrik Abel

Answer

Carl Friedrich Gauss

Answer

Pierre-Simon Laplace

Question 3

What is the order of the cyclic group generated by an element a in G, where a^n = e?

Accepted Answer

n

Answer

m

Answer

mn

Answer

gcd(m, n)

Question 4

Which of the following is NOT a characteristic of a finite group?

Accepted Answer

Its homomorphisms are always surjective

Answer

Its order is finite

Answer

Its subgroups are finite

Answer

Its elements have finite order

Question 5

Which mathematical structure can be generated by a single element?

Accepted Answer

Cyclic group

Answer

Multiplicative group of complex numbers

Answer

Symmetric group on n letters

Answer

Dihedral group of order 8

Question 6

Which definition most accurately describes the order of an element in a group?

Accepted Answer

Smallest positive integer n such that element raised to the power n equals the identity element

Answer

Number of occurrences of the element in the group

Answer

Number of subgroups generated by the element

Answer

Number of homomorphisms from the group to the element

Question 7

Which notation represents the direct product of groups G and H?

Accepted Answer

G x H

Answer

G * H

Answer

G + H

Answer

G - H

Question 8

Which of the following is NOT a significant theorem in group theory?

Accepted Answer

Fermat's little theorem

Answer

Lagrange's theorem

Answer

Cayley's theorem

Answer

Sylow's theorems

Question 9

Which of the following is not an essential property of a group?

Accepted Answer

Commutativity of the operation

Answer

Closure under the operation

Answer

Existence of an identity element

Answer

Associativity of the operation

Question 10

Which group is isomorphic to ℤ/(4) × ℤ/(5)?

Accepted Answer

ℤ/(20)

Answer

ℤ/(4) + ℤ/(5)

Answer

ℤ/(2) × ℤ/(10)

Answer

The symmetric group S₄

Question 11

Find the order of the quotient group S₃/A₃.

Accepted Answer

3

Answer

1

Answer

2

Answer

6

Question 12

Find a generating set for the group ℤ/(12) × ℤ/(8).

Accepted Answer

{(1, 0), (0, 1)}

Answer

{(1, 1), (0, 2)}

Answer

{(2, 0), (0, 3)}

Answer

{(3, 0), (0, 4)}

Question 13

Which of the following is a necessary and sufficient condition for a set to be a group under a given operation?

Accepted Answer

The operation is associative, has an identity element, and every element has an inverse element.

Answer

The operation is associative and commutative.

Answer

The operation has an identity element.

Answer

The operation is associative and has an inverse element for every element.

Question 14

Which of the following properties is not applicable to every cyclic group?

Accepted Answer

It is necessarily abelian.

Answer

It is generated by a single element.

Answer

All of its elements have finite order.

Answer

It has a unique subgroup of each order.

Question 15

Which of the following functions is a homomorphism between the groups (Z_4, +) and (Z_2, +)?

Accepted Answer

f(x) = x mod 2

Answer

f(x) = 2x

Answer

f(x) = x^2

Answer

f(x) = x + 2

Question 16

Which of the following sets is a subgroup of the group of integers under the operation of addition?

Accepted Answer

The set of even integers

Answer

The set of prime numbers

Answer

The set of rational numbers

Answer

The set of complex numbers

Question 17

Which of the following is the definition of a homomorphism between groups?

Accepted Answer

A mapping that maintains the group operation

Answer

A mapping that maintains the identity element

Answer

A mapping that maintains the inverse elements

Answer

A mapping that maintains the order of elements

Question 18

If G is a group and H is a subset of G that forms a group under the same operation as G, then H is called a:

Accepted Answer

Subgroup of G

Answer

Proper subgroup of G

Answer

Factor group of G

Answer

Quotient group of G

Question 19

Which of the following is a homomorphism between groups?

Accepted Answer

A function that preserves the operation and identity element

Answer

A function that preserves the identity element and inverse elements

Answer

A function that preserves the inverse elements and order of elements

Answer

A function that preserves the order of elements and operation

Question 20

Which of the following is an example of a non-abelian group?

Accepted Answer

The permutation group S3

Answer

The group of integers under addition

Answer

The group of rational numbers under multiplication

Answer

The group of complex numbers under multiplication

Question 21

In group theory, explain the significance of simple groups and provide an example.

Accepted Answer

Simple groups are groups that contain only the trivial subgroup and the group itself as normal subgroups. They are crucial for understanding the structure of finite groups.

Answer

Simple groups are characterized by having unique elements, providing insights into the composition of groups.

Answer

Simple groups lack any subgroups, making them the most basic type of group.

Answer

Simple groups have only one operation, offering a fundamental understanding of group theory.

Question 22

Which of the following is a limitation of group theory?

Accepted Answer

Inability to deal with continuous symmetries

Answer

Inability to represent finite sets

Answer

Inability to handle non-commutative operations

Answer

Requirement that all elements have an inverse

Question 23

Which of the following is NOT a defining property of a group?

Accepted Answer

Closure under division

Answer

Existence of inverses

Answer

Closure under the operation

Answer

Existence of an identity element

Question 24

What is the order of the group (Z_5, +)?

Accepted Answer

5

Answer

6

Answer

4

Answer

10

Question 25

What is the cyclic group generated by the element 2 in the group (Z_7, +)?

Accepted Answer

{0, 2, 4, 6}

Answer

{1, 2, 3, 4, 5, 6}

Answer

{0, 1, 2, 3, 4, 5, 6}

Answer

{2, 4, 6}

Question 26

Which of the following is NOT a homomorphism from the group (Z, +) to the group (Z_4, +)?

Accepted Answer

f(x) = x^2

Answer

f(x) = 0

Answer

f(x) = x mod 4

Answer

f(x) = 2x mod 4

Question 27

What is the kernel of the homomorphism f: (Z, +) -> (Z_4, +) defined by f(x) = x mod 4?

Accepted Answer

4Z

Answer

Z

Answer

{0}

Answer

2Z

Question 28

What is the order of the symmetric group S_4?

Accepted Answer

24

Answer

16

Answer

4

Answer

12

Question 29

Which of the following is NOT a subgroup of the group (Z, +)?

Accepted Answer

{1, 2, 3}

Answer

{0, 2, 4, 6}

Answer

{..., -6, -3, 0, 3, 6, ...}

Answer

{0, 1, 2, 3}

Question 30

What is the isomorphism class of a cyclic group of order 6?

Accepted Answer

Z_6

Answer

S_3

Answer

Z_4

Answer

Z_2 x Z_3

Question 31

Which of the following is NOT a property of a group?

Accepted Answer

The operation is commutative

Answer

There exists an identity element

Answer

The operation is associative

Answer

Every element has an inverse

Question 32

Given the group (Z, +), what is the order of the element 3?

Accepted Answer

Infinite

Answer

1

Answer

3

Answer

6

Question 33

Given a group G with subgroups H and K, which of the following statements is always true?

Accepted Answer

The intersection of H and K (H ∩ K) is a subgroup of G.

Answer

H is a subset of K or K is a subset of H (H ⊆ K or K ⊆ H).

Answer

The union of H and K (H ∪ K) is a subgroup of G.

Question 34

Let φ be a group homomorphism from group G to group H. Which statement is always true about φ?

Accepted Answer

The image of the identity element of G (e_G) under φ is the identity element of H (e_H): φ(e_G) = e_H.

Answer

φ is a bijective function (both injective and surjective).

Answer

φ maps every element of G to itself: φ(x) = x for all x in G.

Question 35

Which of the following sets, equipped with the specified operation, forms a cyclic group?

Accepted Answer

The set of integers modulo 5 (Z_5) under addition (+).

Answer

The set of rational numbers (Q) under addition (+).

Answer

The general linear group of 2x2 matrices with real entries (GL(2, R)) under matrix multiplication (×).

Answer

The symmetric group of degree 3 (S_3) under composition (◦).

Question 36

What is the order of the permutation (1 2 3)(4 5)?

Accepted Answer

6

Answer

5

Answer

3

Answer

2

Question 37

A group G is considered simple if it satisfies which of the following conditions?

Accepted Answer

G does not contain any proper nontrivial normal subgroups.

Answer

G is isomorphic to a subgroup of a symmetric group.

Answer

G can be generated by a single element (cyclic group).

Answer

G has a finite number of elements (finite order).

Question 38

In a group homomorphism φ from G to H, the kernel of φ is defined as:

Accepted Answer

The set of elements in G that map to the identity element of H: {g ∈ G | φ(g) = e_H}.

Answer

The set of elements in G that map to themselves: {g ∈ G | φ(g) = g}.

Answer

The set of elements in H that are the image of some element in G: {h ∈ H | φ(g) = h for some g ∈ G}.

Answer

The set of elements in G that map to the identity element of G: {g ∈ G | φ(g) = e_G}.

Question 39

Which of the following functions is an example of a group isomorphism?

Accepted Answer

φ: (Z_4, +) → (Z_4, +) defined by φ(x) = 2x (mod 4).

Answer

φ: (Z, +) → (Q, +) defined by φ(x) = x/2.

Answer

φ: (R, +) → (R, ×) defined by φ(x) = e^x.

Question 40

Lagrange's theorem in group theory states that:

Accepted Answer

The order of any subgroup of a finite group is a divisor of the order of the group.

Answer

Every group has a unique element that acts as the identity.

Answer

Every group with a finite number of elements can be generated by a single element.

Question 41

Consider the group (Z, +). What is the order of the element 3?

Accepted Answer

Infinite

Answer

3

Answer

6

Answer

1

Question 42

Let G be a group and H be a subgroup of G. Which of the following is always true?

Accepted Answer

The identity element of G is also the identity element of H.

Answer

H is a cyclic group.

Answer

The order of H is equal to the order of G.

Question 43

Consider the homomorphism f: (Z, +) -> (Z_4, +) defined by f(x) = x mod 4. What is the kernel of f?

Accepted Answer

{..., -4, 0, 4, 8, ...}

Answer

{0, 2}

Answer

{0, 1, 2, 3}

Answer

{1, 3}

Question 44

Which of the following groups is cyclic?

Accepted Answer

(Z_5, +)

Answer

(Z_6, +)

Answer

(S_3, o)

Answer

(Q, *)

Question 45

Let G be a group and a, b be elements of G. Which of the following is always true about the commutator [a, b] = a⁻¹b⁻¹ab?

Accepted Answer

[a, b] = e if and only if a and b commute.

Answer

[a, b] = ab⁻¹.

Answer

[a, b] = a²b².

Answer

[a, b] = a⁻¹b.

Question 46

Consider the permutation (1 2 3)(4 5) in S_5. What is its order?

Accepted Answer

6

Answer

2

Answer

5

Answer

3

Question 47

Which of the following is a normal subgroup of the group (Z_6, +)?

Accepted Answer

{0, 3}

Answer

{0, 4}

Answer

{0, 2}

Answer

{0, 1}

Question 48

What is the index of the subgroup H = {e, (1 2)} in the symmetric group S_3?

Accepted Answer

3

Answer

6

Answer

2

Answer

4

Question 49

Which of the following statements about Lagrange's Theorem is TRUE?

Accepted Answer

The order of a subgroup divides the order of the group.

Answer

The order of a group is always equal to the sum of the orders of its subgroups.

Answer

Every group has a subgroup of order 2.