Groups: Definition and Examples - Abstract Algebra Quizzes

Question 1

Which of these is NOT a requirement for a set to be considered a group?

Accepted Answer

The operation is commutative.

Answer

The operation is associative.

Answer

There is an identity element.

Answer

Every element has an inverse.

Question 2

What is the precise definition of a group?

Accepted Answer

A set G with an operation ∘ that combines any elements a and b in G to form a unique element a ∘ b in G.

Answer

A collection of objects that can be combined in a specific way to produce a new object in the collection.

Answer

A set of objects that have an identity element and an inverse element for each object.

Answer

A set of objects that satisfy the associative, identity, and inverse properties.

Question 3

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

Accepted Answer

The set of all integers under the operation of addition

Answer

The set of all real numbers under the operation of multiplication

Answer

The set of all complex numbers under the operation of addition

Answer

The set of all matrices under the operation of matrix multiplication

Question 4

Which of the following properties is NOT true for all groups?

Accepted Answer

Commutative property

Answer

Associative property

Answer

Identity element

Answer

Inverse element

Question 5

What is the identity element of the group of integers under the operation of addition?

Accepted Answer

0

Answer

1

Answer

2

Answer

3

Question 6

Which of the following is a subgroup of the symmetric group S₄?

Accepted Answer

The set of all even permutations

Answer

The set of all odd permutations

Answer

The set of all permutations that fix a particular element

Answer

The set of all permutations that cycle three elements

Question 7

Which of the following is a homomorphism between the groups (Z₆, +) and (Z₂, +)?

Accepted Answer

f(x) = x mod 2

Answer

f(x) = x mod 3

Answer

f(x) = x mod 4

Answer

f(x) = x mod 5

Question 8

Which property is not always true for groups?

Accepted Answer

Commutativity of the group operation

Answer

Closure under the group operation

Answer

Associativity of the group operation

Answer

Existence of an identity element

Question 9

Which of the following sets, when equipped with the specified binary operation, fails to form a group?

Accepted Answer

Real numbers under subtraction

Answer

Integers under addition

Answer

Rational numbers under multiplication

Answer

Non-zero real numbers under multiplication

Question 10

Which of the following defines a group?

Accepted Answer

A set with an operation satisfying associativity, inverse, and identity properties

Answer

A set with an operation satisfying associativity, commutativity, and identity properties

Answer

A set with an operation satisfying associativity, closure, and identity properties

Answer

A set with an operation satisfying commutativity, inverse, and closure properties

Question 11

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

Accepted Answer

The set of integers under addition

Answer

The set of real numbers under multiplication

Answer

The symmetric group on three elements

Answer

The dihedral group of order 8

Question 12

What is the order of the symmetric group on n elements?

Accepted Answer

n!

Answer

n

Answer

n^2

Answer

2^n

Question 13

What is the center of a group G?

Accepted Answer

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

Answer

The set of elements in G that are equal to their own inverses

Answer

The set of elements in G that generate the group

Answer

The set of elements in G that have order 2

Question 14

Which of the following is a homomorphism?

Accepted Answer

A function between two groups that preserves the group operation

Answer

A function between two groups that preserves the identity element

Answer

A function between two groups that preserves the inverse operation

Answer

A function between two groups that preserves all of the above

Question 15

What is the kernel of a homomorphism between two groups?

Accepted Answer

The set of elements in the domain that map to the identity element in the codomain

Answer

The set of elements in the codomain that are not in the image of the homomorphism

Answer

The set of elements in the domain that are in the image of the homomorphism

Answer

The set of elements in the codomain that have order 2

Question 16

Consider the operation defined on the set {a, b, c} as follows:
a * b = c
b * c = a
c * a = d
Does this operation satisfy the closure property?

Accepted Answer

No, the operation does not satisfy the closure property.

Answer

Yes, the operation satisfies the closure property.

Answer

The operation satisfies the closure property only for certain elements.

Question 17

Identify an example of a cyclic group:

Accepted Answer

The set of integers under the operation of addition

Answer

The set of rational numbers under the operation of multiplication

Answer

The set of complex numbers under the operation of addition

Answer

The set of real numbers under the operation of multiplication

Question 18

Determine the order of the symmetric group S3:

Accepted Answer

6

Answer

3

Answer

9

Answer

12

Question 19

Provide a definition for a homomorphism:

Accepted Answer

A function between two groups that preserves the group operation.

Answer

A function between two sets that preserves the binary operation.

Answer

A function between two groups that preserves the identity element.

Answer

A function between two sets that preserves the inverse operation.

Question 20

State a consequence of Lagrange's theorem:

Accepted Answer

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

Answer

The order of any quotient group of a group divides the order of the group.

Answer

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

Answer

The order of a group is equal to the order of its quotient groups.

Question 21

Which characteristic accurately describes a finite group?

Accepted Answer

It has a finite number of elements.

Answer

Its order is necessarily a prime number.

Answer

Its center is always trivial.

Answer

Its subgroups are all cyclic.

Question 22

Identify a property of nilpotent groups:

Accepted Answer

They have a chain of subgroups such that each subgroup is a normal subgroup of the next.

Answer

They have a solvable series of subgroups.

Answer

Their center is equal to the entire group.

Answer

They are all finite groups.

Question 23

According to Lagrange's theorem, which statement is accurate?

Accepted Answer

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

Answer

The order of a group is always a multiple of 2.

Answer

Any group with an odd number of elements is cyclic.

Question 24

Which of the following sets does NOT form a group under function composition?

Accepted Answer

All real numbers under addition

Answer

All permutations of a finite set

Answer

All invertible square matrices of a given size

Answer

All complex numbers under multiplication

Question 25

Which of the following is an illustration of a cyclic group?

Accepted Answer

The set of integers under the operation of addition

Answer

The set of complex numbers under the operation of division

Answer

The set of rational numbers under the operation of multiplication

Answer

The set of real numbers under the operation of subtraction

Question 26

Which of the following exemplifies a symmetric group?

Accepted Answer

The set of all permutations of 3 elements

Answer

The set of rational numbers under the operation of multiplication

Answer

The set of integers under the operation of addition

Answer

The set of real numbers under the operation of subtraction

Question 27

Which property is NOT characteristic of a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Inverse element

Answer

Identity element

Question 28

Which set is a subgroup of the group of integers under addition?

Accepted Answer

The set of even integers

Answer

The set of negative integers

Answer

The set of prime integers

Question 29

Which function represents an isomorphism between the group of integers under addition and the group of even integers under addition?

Accepted Answer

f(x) = 2x

Answer

f(x) = x^2

Answer

f(x) = x + 1

Question 30

What is the order of the group of permutations of 3 elements?

Accepted Answer

6

Answer

9

Answer

12

Answer

3

Question 31

Which set forms the center of the group of integers under addition?

Accepted Answer

The set of multiples of 0

Answer

The set of prime integers

Answer

The set of even integers

Answer

The set of negative integers

Question 32

Which of the following is a property of groups?

Accepted Answer

The associative property holds for all elements.

Answer

The inverse element exists for some elements.

Answer

The identity element does not exist.

Question 33

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

Accepted Answer

The set of integers under addition.

Answer

The set of quaternions under multiplication.

Answer

The set of rational numbers under multiplication.

Answer

The set of complex numbers under multiplication.

Question 34

Consider the set S = {1, 2, 3} under the operation of composition of permutations. Is S a group?

Accepted Answer

Yes, S is a group.

Answer

No, S is not a group because the inverse element does not exist for every element.

Answer

No, S is not a group because the identity element does not exist.

Answer

No, S is not a group because the associative property does not hold for all elements.

Question 35

Which of the following is an example of a finite group?

Accepted Answer

The set of invertible 2x2 matrices with integer entries.

Answer

The set of all rational numbers under addition.

Answer

The set of all real numbers under multiplication.

Question 36

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

Accepted Answer

The multiplicative group of 2x2 matrices with real entries.

Answer

The group of integers under addition modulo 5.

Answer

The additive group of real numbers.

Question 37

Which of the following best defines a group?

Accepted Answer

A non-empty set G together with an operation * on G that satisfies the associativity, identity, and inverse properties

Answer

A non-empty set with an identity element

Answer

A set with an operation that satisfies the closure property

Question 38

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

Accepted Answer

Commutativity

Answer

Inverse element

Answer

Associativity

Answer

Identity element

Question 39

What is the identity element of the group <Z, +>?

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 40

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

Accepted Answer

<S3, o>

Answer

<Q, +>

Answer

<Z, +>

Question 41

Which of the following operations on the set of real numbers is NOT closed?

Accepted Answer

Division

Answer

Multiplication

Answer

Subtraction

Answer

Addition

Question 42

In the group of invertible 2x2 matrices with real entries under matrix multiplication, which of the following is the identity element?

Accepted Answer

The 2x2 identity matrix

Answer

A matrix with -1s on the diagonal and 0s elsewhere

Answer

A matrix with 1s on the diagonal and 0s elsewhere

Answer

The zero matrix

Question 43

Which of the following sets forms a cyclic group under multiplication?

Accepted Answer

The set of complex numbers of the form z^n, where z is a fixed complex number and n is an integer

Answer

The set of all positive integers

Answer

The set of all real numbers

Question 44

The symmetric group S_3 consists of all permutations of the set {1, 2, 3}. What is the order of the group S_3?

Accepted Answer

6

Answer

9

Answer

12

Answer

3

Question 45

Which of the following concepts is central to the unit on Definition and Examples?

Accepted Answer

Groups

Answer

Vector Spaces

Answer

Rings

Answer

Linear Algebra

Question 46

Cyclic groups are a type of:

Accepted Answer

Group

Answer

Field

Answer

Algebra

Answer

Ring

Question 47

The symmetric group S3 contains how many distinct elements?

Accepted Answer

6

Answer

3

Answer

24

Answer

12

Question 48

For a binary operation on a set to be well-defined, it must satisfy which of the following conditions?

Accepted Answer

The operation must be defined for any pair of elements in the set.

Answer

The operation must be associative.

Answer

The operation must have an identity element.

Question 49

Consider the set {0, 1} with the operation ⊕ defined as: a ⊕ b = 1 if a ≠ b and a ⊕ b = 0 if a = b. Is this operation commutative?

Accepted Answer

Yes

Answer

Only commutative for a = 0 and b = 1

Answer

Cannot be determined

Answer

No

Question 50

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

Accepted Answer

The group of invertible 2 x 2 matrices under matrix multiplication

Answer

The group of complex numbers under multiplication

Answer

The group of integers under addition

Question 51

In the group (Z, +), what is the inverse of -5?

Accepted Answer

5

Answer

-1

Answer

0

Answer

Cannot be determined

Question 52

Which of the following is the definition of a trivial group?

Accepted Answer

A group with only one element

Answer

A group with infinitely many elements

Answer

A group with a finite number of elements greater than 1

Answer

A group with an even number of elements

Question 53

The order of an element 'a' in a group is defined as the smallest positive integer 'n' such that:

Accepted Answer

a^n = e, where e is the identity element

Answer

a^-n = e

Answer

a^n = 0

Answer

a^n = a