Master Group Theory Concepts: Online Quizzes for Engineering Mathematics

Question 1

In group theory, define a subgroup and explain its significance.

Accepted Answer

A subgroup is a subset of a group that inherits the group operation and satisfies the group axioms within its subset.

Answer

A subgroup is a strictly larger group that contains the original group.

Answer

A subgroup is a set of elements that commutes with all elements of the original group.

Answer

A subgroup is a set of elements that generate the entire original group.

Question 2

How are group theory and ring theory related, and in which engineering applications is this relationship utilized?

Accepted Answer

Group theory and ring theory are related algebraic structures used in coding theory, computer algebra, and cryptography.

Answer

Group theory and ring theory are unrelated mathematical concepts with no practical engineering applications.

Answer

Group theory is a branch of ring theory, and they are used interchangeably in engineering applications.

Answer

The relationship between group theory and ring theory is purely theoretical, with no practical engineering applications.

Question 3

Question: Explain the relationship between group theory and abstract algebra. How does group theory enhance our understanding of algebraic structures within the broader framework of abstract algebra?

Accepted Answer

Group theory is a fundamental component of abstract algebra, focusing on groups with distinct algebraic properties.

Answer

Group theory is an independent field, unrelated to abstract algebra.

Answer

Abstract algebra is a subset of group theory, studying general features of algebraic structures, including groups.

Answer

Group theory and abstract algebra are separate disciplines that share only terminology.

Question 4

Which of the following is a property of a finite group?

Accepted Answer

Every element has finite order

Answer

The order of the group is infinite

Answer

The group is not cyclic

Answer

The group is Abelian

Question 5

Consider the permutation group S₃, which is the group of permutations of {1, 2, 3}. What is the identity element of S₃?

Accepted Answer

(1 2 3)

Answer

(1 3)

Answer

(1 3 2)

Answer

(1 2)

Question 6

Precise Question: Which of the following is NOT a defining property of a group?

Accepted Answer

The operation is commutative (the order of elements does not affect the result).

Answer

The operation combines any two elements to produce another element in the group (closure).

Answer

The operation is associative (the grouping of elements does not affect the result).

Answer

There exists an identity element that leaves any element unchanged when combined with it.

Question 7

Precise Question: Which of the following functions is a homomorphism from the group (Z, +) to the group (R, +)?

Accepted Answer

f(x) = 2x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 8

Precise Question: The symmetry group of a square has order:

Accepted Answer

8

Answer

2

Answer

4

Answer

16

Question 9

Precise Question: Which of the following crystal systems exhibits the highest symmetry?

Accepted Answer

Cubic

Answer

Tetragonal

Answer

Orthorhombic

Answer

Monoclinic

Question 10

Precise Question: Which of the following is a normal subgroup of the symmetric group on n letters?

Accepted Answer

The alternating group on n letters

Answer

The group of all permutations on n letters

Answer

The group of all odd permutations on n letters

Answer

The group of all even permutations on n letters

Question 11

Precise Question: The centre of a group is defined as:

Accepted Answer

The set of all elements that commute with every other element in the group.

Answer

The set of all elements that commute with the identity element of the group.

Answer

The set of all elements that are their own inverses.

Answer

The set of all elements that have order 2.

Question 12

Precise Question: Which of the following is a direct consequence of Lagrange's theorem?

Accepted Answer

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

Answer

The order of a group is always even.

Answer

The group is cyclic.

Answer

The group is abelian.

Question 13

Precise Question: The group of symmetries of a regular n-gon is isomorphic to:

Accepted Answer

The dihedral group of order 2n

Answer

The cyclic group of order n

Answer

The alternating group on n letters

Answer

The symmetric group on n letters

Question 14

Which property is NOT characteristic of a group?

Accepted Answer

Commutativity of the operation

Answer

Associativity of the operation

Answer

Closure under the operation

Answer

Existence of an identity element

Question 15

Let H be a subgroup of a group G. Which statement is always true about the order of H?

Accepted Answer

It is a factor of the order of G

Answer

It can be greater than the order of G

Answer

It is equal to the order of G

Answer

It is always even

Question 16

Which of the following groups is cyclic?

Accepted Answer

Z (the integers under addition)

Answer

Q (the rational numbers under addition)

Answer

S3 (the symmetric group on three elements)

Answer

R (the real numbers under addition)

Question 17

Which of the given functions qualifies as a homomorphism between the groups (Z, +) and (R, +)?

Accepted Answer

f(x) = 2x

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = e^x

Question 18

What is the order of the group (S3, ∘)?

Accepted Answer

6

Answer

3

Answer

4

Answer

8

Question 19

Which of the following statements accurately describes a finite group?

Accepted Answer

It has a finite number of elements

Answer

It can only have cyclic subgroups

Answer

It is always Abelian

Answer

It cannot have any subgroups

Question 20

Which of the following is a direct consequence of Sylow's Theorem?

Accepted Answer

Every group has at least one normal subgroup of prime order

Answer

Every finite group has a unique subgroup of the highest order

Answer

Every finite group is solvable

Answer

Every finite group is simple

Question 21

Which property is NOT required for a set to form a group under a given operation?

Accepted Answer

Commutativity

Answer

Closure

Answer

Associativity

Answer

Identity element

Question 22

In the cyclic group Z_n, what is the order of the element 'a'?

Accepted Answer

gcd(a, n)

Answer

n

Answer

lcm(a, n)

Answer

a

Question 23

What is the number of elements in the dihedral group D_n?

Accepted Answer

2n

Answer

n

Answer

2n^2

Answer

n^2

Question 24

What is the definition of the center of a group G?

Accepted Answer

The set of elements that commute with every other element in G

Answer

The set of elements that are centralized by every other element in G

Answer

The intersection of all subgroups of G

Answer

The union of all subgroups of G

Question 25

Within the cyclic group Z_5 (addition modulo 5), what is the order of the element 3?

Accepted Answer

3

Answer

1

Answer

2

Answer

5

Question 26

Which of these is an example of a non-abelian group?

Accepted Answer

(S_3, ∘)

Answer

(Z, +)

Answer

(R, ×)

Answer

(Q, ×)

Question 27

Is the statement 'Every subgroup of a finite group is finite' true or false?

Accepted Answer

True

Answer

False

Question 28

Within the group G = {1, -1, i, -i} under multiplication, which element acts as a generator?

Accepted Answer

i

Answer

1

Answer

-1

Question 29

Which application of group theory involves the study of symmetry in crystals?

Accepted Answer

Symmetry analysis

Answer

Error correction

Answer

Data compression

Answer

Cryptography

Question 30

Which of the following defines a homomorphism between groups?

Accepted Answer

A map that preserves the group operation

Answer

A map that preserves the identity element

Answer

A map that preserves the inverse operation

Answer

A map that preserves the order of elements

Question 31

Which property is NOT essential for a set to be considered a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Answer

Closure

Question 32

Which of the following functions between groups G and H is a homomorphism?

Accepted Answer

A function that preserves the group operation.

Answer

A function that preserves the identity element.

Answer

A function that is injective.

Answer

A function that is surjective.

Question 33

What is the symmetry group of a square?

Accepted Answer

D_4

Answer

C_4

Answer

S_4

Answer

A_4

Question 34

What is the centre of a group?

Accepted Answer

The set of all elements that commute with every other element.

Answer

The set of all elements that are their own inverses.

Answer

The set of all elements that generate the group.

Answer

The set of all elements that have order 2.

Question 35

Which of the following is a defining property of a normal subgroup?

Accepted Answer

It is invariant under conjugation.

Answer

It is generated by a single element.

Answer

It is the kernel of a homomorphism.

Answer

It is isomorphic to the parent group.

Question 36

What does the quotient group G/H represent?

Accepted Answer

The set of all left cosets of H in G.

Answer

The set of all right cosets of H in G.

Answer

The set of all subgroups of H.

Answer

The set of all homomorphisms from G to H.

Question 37

Which property is NOT characteristic of a group?

Accepted Answer

Commutative operation

Answer

Closure under the operation

Answer

Associative operation

Answer

Existence of an identity element

Question 38

Which among the following is an example of a cyclic group?

Accepted Answer

Group of integers under addition

Answer

Group of complex numbers under multiplication

Answer

Symmetric group on 3 elements

Answer

Dihedral group of order 8

Question 39

The group of rotations of a regular hexagon is isomorphic to which group?

Accepted Answer

Dihedral group of order 6

Answer

Cyclic group of order 6

Answer

Symmetric group on 3 elements

Answer

Alternating group on 4 elements

Question 40

Identify an example of a cyclic group:

Accepted Answer

The set of integers under the addition operation

Answer

The set of non-zero rational numbers under multiplication

Answer

The set of symmetries of a square

Answer

The set of permutations of three elements

Question 41

Within the group (Z, +), which element acts as the identity?

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 42

What is the order of the group generated by the element (1 2) in the symmetric group S3?

Accepted Answer

3

Answer

1

Answer

2

Answer

6

Question 43

Which of the following is a fundamental theorem in group theory?

Accepted Answer

Lagrange's theorem

Answer

The fundamental theorem of algebra

Answer

The Pythagorean theorem

Answer

The binomial theorem

Question 44

What is the minimum order of a non-abelian group?

Accepted Answer

3

Answer

2

Answer

4

Answer

6

Question 45

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

Accepted Answer

Existence of a multiplicative inverse for every element

Answer

Closure under the group operation

Answer

Associativity of the group operation

Answer

Existence of an identity element

Question 46

Let $H$ be a subset of a group $G$. Which of the following is a necessary and sufficient condition for $H$ to be a subgroup of $G$?

Accepted Answer

Closure under the group operation and containing the identity element

Answer

Closure under the group operation and non-empty

Answer

Closed under the inverse operation and containing the identity element

Answer

Closed under the conjugation operation and containing the identity element

Question 47

Which of the following is a homomorphism between groups $G$ and $H$?

Accepted Answer

A function that preserves both the group operation and the identity element

Answer

A function that preserves the group operation but not the identity element

Answer

A function that preserves the identity element but not the group operation

Answer

A function that does not preserve either the group operation or the identity element

Question 48

The dihedral group $D_4$ has how many elements?

Accepted Answer

8

Answer

2

Answer

4

Answer

16

Question 49

What is a free product of groups, and why is it significant in group theory?

Accepted Answer

A free product of groups is a new group formed by combining existing groups such that elements from different groups commute.

Answer

A free product of groups is obtained by multiplying two or more groups together.

Answer

A free product of groups results in a group that is smaller than the original groups.

Answer

Free products of groups are trivial groups.

Question 50

What is the order of the cyclic group generated by the element (1 2 3)?

Accepted Answer

6

Answer

1

Answer

2

Answer

3

Question 51

What is the kernel of the homomorphism f: Z₆ → Z₂ given by f(x) = x mod 2?

Accepted Answer

{0, 2, 4}

Answer

{1, 3, 5}

Answer

{0, 3}

Answer

{1, 4}

Question 52

Which of the following statements about the center of a group is true?

Accepted Answer

It is the set of all elements that commute with all other elements of the group.

Answer

It is a normal subgroup of the group.

Answer

It is always isomorphic to a cyclic group.

Answer

It is the smallest subgroup of the group.

Question 53

What is Lagrange's theorem?

Accepted Answer

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

Answer

The number of elements in a group is always a power of 2.

Answer

The inverse of every element in a group is unique.

Answer

The identity element is the only element that commutes with all other elements in a group.

Question 54

If G is a group and H is a non-empty subset of G that is closed under the group operation, then H is a:

Accepted Answer

Subgroup of G

Answer

Normal subgroup of G

Answer

Homomorphism of G

Answer

Isomorphism of G

Question 55

Which of the following is a necessary and sufficient condition for a group to be cyclic?

Accepted Answer

There exists an element in the group that generates the entire group

Answer

Every element in the group has finite order

Answer

The group is abelian

Answer

The group has a unique identity element

Question 56

What is the order of the Klein 4-group?

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 57

Which of the following is a characteristic of a finite group?

Accepted Answer

Every element has finite order

Answer

Every subgroup is finite

Answer

The order of the group is always even

Answer

The group is always abelian

Question 58

Which of the following is **not** a defining property of a group?

Accepted Answer

Commutativity of the operation

Answer

Associativity of the operation

Answer

Existence of an identity element

Answer

Existence of an inverse element for each element

Question 59

Consider the set of all rotations of a square. Which of the following is **not** a subgroup of this group?

Accepted Answer

The set of translations

Answer

The set of rotations by 90 degrees

Answer

The set of reflections about the diagonals

Answer

The set of rotations by 60 degrees

Question 60

Which of the following is **not** an application of group theory in engineering?

Accepted Answer

Fluid dynamics

Answer

Symmetry analysis of structures

Answer

Crystallography

Answer

Design of error-detecting codes

Question 61

Let (G) be a group and let (a, b) be elements of (G). Which of the following is equivalent to the statement "a commutes with b"?

Accepted Answer

$$ab = ba$$

Answer

$$a^{-1}b = ba^{-1}$$

Answer

$$a^2b = b^2a$$

Answer

$$a^{-1}b^{-1} = b^{-1}a^{-1}$$

Question 62

Define a group. Your definition should clearly state the properties of a group.

Accepted Answer

A set of elements with an operation that combines any two elements to form a third element in the set, an identity element, and an inverse element for each element.

Answer

A set of elements with an operation that combines any two elements to form a unique element in the set, and an identity element.

Answer

A set of elements with an operation that combines any two elements to form an element that is not in the set.

Answer

A set of elements with an operation that does not combine any two elements.

Question 63

What is the order of the group S3 (the group of permutations on three elements)?

Accepted Answer

6

Answer

3

Answer

9

Answer

12

Question 64

State a fundamental theorem of group theory.

Accepted Answer

Every abelian group is isomorphic to a direct product of cyclic groups

Answer

Every group is isomorphic to a subgroup of a permutation group

Answer

Every finite group is a subgroup of a finite simple group

Answer

Every non-abelian group is isomorphic to a subgroup of a non-abelian simple group