Examples of Groups in Mathematics: Interactive Learning Quizzes

Question 1

Identify the set that does NOT form a group under the specified operation:

Accepted Answer

Complex numbers under subtraction

Answer

Integers under addition

Answer

Rational numbers under multiplication

Question 2

Identify the example of a cyclic group:

Accepted Answer

Complex numbers of unit magnitude under multiplication

Answer

Rational numbers under addition

Answer

Integers under subtraction

Question 3

In the group of rational numbers excluding 0 under multiplication, the inverse of -3 is:

Accepted Answer

1/3

Answer

-3

Answer

3

Answer

-1/3

Question 4

Identify the identity element of the integers (ℤ) under addition.

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 5

Which property is NOT satisfied by a group?

Accepted Answer

Commutativity

Answer

Closure

Answer

Associativity

Answer

Existence of both identity and inverse elements

Question 6

Which of the following is **not** an example of a group under the given operation?

Accepted Answer

Natural numbers under subtraction

Answer

Integers under addition

Answer

Rational numbers under multiplication

Answer

Complex numbers under addition

Question 7

In a finite group G with order 12, subgroup H has order 3. What's the index of H in G?

Accepted Answer

4

Answer

3

Answer

6

Question 8

Consider the integers under addition and the group of rational numbers excluding 0 under multiplication. Determine if the map f(x) = 2x from the integers to the rational numbers is a homomorphism.

Accepted Answer

Yes, because f(x + y) = f(x) + f(y) for all integers x and y.

Answer

No, because f(xy) = f(x) * f(y) for all integers x and y.

Answer

The integers are not closed under multiplication.

Question 9

The set of all 2x2 matrices with real entries and determinant 1 forms a group under matrix multiplication. Which of the following is the identity element of this group?

Accepted Answer

The matrix [[1, 0], [0, 1]]

Answer

The matrix [[0, 1], [1, 0]]

Answer

The matrix [[2, 1], [1, 2]]

Answer

The matrix [[1, 2], [3, 4]]

Question 10

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

Accepted Answer

The set of integers modulo 5 under addition

Answer

The set of rational numbers under multiplication

Answer

The set of complex numbers under multiplication

Answer

The set of real numbers under addition

Question 11

The set of all permutations of a set with n elements forms a group under the operation of composition. What is the order of this group?

Accepted Answer

n!

Answer

n

Answer

n^2

Answer

2n

Question 12

What is the identity element of the group of complex numbers under addition?

Accepted Answer

0

Answer

1

Answer

i

Answer

-1

Question 13

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

Accepted Answer

Closure under multiplication

Answer

Inverses

Answer

Associativity

Answer

Identity element

Question 14

What is the order of the group of rational numbers under multiplication?

Accepted Answer

Infinite

Answer

1

Answer

3

Answer

2

Question 15

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

Accepted Answer

Even numbers

Answer

Odd numbers

Answer

Prime numbers

Answer

Negative numbers

Question 16

What is the inverse of the element 3 in the group of integers under addition?

Accepted Answer

-3

Answer

0

Answer

2

Answer

1

Question 17

What is the group of symmetries of a square?

Accepted Answer

Dihedral group of order 8

Answer

Symmetric group of order 4

Answer

Cyclic group of order 4

Answer

Alternating group of order 4

Question 18

Which of the following is not a property of a cyclic group?

Accepted Answer

Every element is its own inverse

Answer

The group is abelian

Answer

The order of the group is equal to the order of the generator

Answer

There is a unique generator

Question 19

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

Accepted Answer

Distributivity over another operation

Answer

Existence of a neutral element

Answer

Existence of an inverse element

Answer

Associativity

Question 20

The order of the group of complex numbers under addition is:

Accepted Answer

Infinite

Answer

2

Answer

3

Answer

4

Question 21

The group of permutations of 3 elements has an order of:

Accepted Answer

6

Answer

4

Answer

3

Answer

5

Question 22

Which of the following sets is not closed under the operation of multiplication?

Accepted Answer

Set of positive integers

Answer

Set of complex numbers

Answer

Set of rational numbers

Question 23

The identity element in the group of real numbers under addition is:

Accepted Answer

0

Answer

-1

Answer

1

Answer

2

Question 24

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

Accepted Answer

Integers under addition modulo n, for any positive integer n

Answer

Complex numbers under addition

Answer

Rational numbers under multiplication

Answer

Symmetric group on 4 elements

Question 25

Which of the following is the identity element of the group of rational numbers under multiplication?

Accepted Answer

1

Answer

-1

Answer

2

Answer

0

Question 26

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

Accepted Answer

Symmetric group on 4 elements

Answer

Group of units in the ring of integers modulo 5

Answer

Quaternion group

Answer

Group of integers under addition

Question 27

The set of complex numbers under the operation of multiplication forms a group because:

Accepted Answer

It satisfies the closure, associativity, identity, and inverse properties.

Answer

Multiplication is a binary operation on the set of complex numbers.

Answer

The set of complex numbers is a non-empty set.

Question 28

The order of a group is defined as:

Accepted Answer

The number of elements in the group.

Answer

The number of generators in the group.

Answer

The number of operations in the group.

Answer

The number of subgroups in the group.

Question 29

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

Accepted Answer

The group of integers under the operation of addition

Answer

The group of rational numbers under the operation of multiplication

Answer

The group of complex numbers under the operation of addition

Answer

The group of positive real numbers under the operation of multiplication

Question 30

The inverse of an element in a group is:

Accepted Answer

The element that, when combined with the given element, results in the identity element.

Answer

The element that, when combined with the given element, results in zero.

Answer

The element that is the same as the given element.

Question 31

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

Accepted Answer

The group of 2x2 matrices with real entries and determinant 1 under matrix multiplication

Answer

The group of rational numbers under the operation of multiplication

Answer

The group of integers under the operation of addition

Question 32

The center of a group is:

Accepted Answer

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

Answer

The set of all elements that generate the group.

Answer

The set of all elements that are equal to their own inverse.

Question 33

Which of the following sets is NOT a group under the operation of addition?

Accepted Answer

The set of real numbers

Answer

The set of integers

Answer

The set of complex numbers

Answer

The set of rational numbers

Question 34

Which of the following sets is a group under the operation of multiplication?

Accepted Answer

The set of rational numbers excluding 0

Answer

The set of real numbers

Answer

The set of negative integers

Answer

The set of odd integers

Question 35

Consider the set G = {1, -1, i, -i} under the operation of multiplication. Which element in G is the identity element?

Accepted Answer

1

Answer

-i

Answer

i

Answer

-1

Question 36

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

Accepted Answer

The set of integers under addition modulo 5

Answer

The set of rational numbers under multiplication

Answer

The set of real numbers under multiplication

Answer

The set of complex numbers under addition

Question 37

Determine which set forms a group under the specified operation:

Accepted Answer

Integers under addition

Answer

Complex numbers under division

Answer

Rational numbers under subtraction

Answer

Irrational numbers under multiplication

Question 38

In the group of complex numbers under addition, find the inverse of (2 + 3i):

Accepted Answer

(2 - 3i)

Answer

(2 + 3i)^-1

Answer

(-2 + 3i)

Answer

(2 - 3i)^-1

Question 39

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

Accepted Answer

Integers under addition

Answer

Symmetric group on 3 elements

Answer

Rational numbers under multiplication

Answer

Complex numbers under addition

Question 40

Determine the number of elements in the group of symmetries of a square:

Accepted Answer

8

Answer

4

Answer

6

Answer

12

Question 41

Identify the operation under which the set of invertible 2x2 matrices with real entries forms a group:

Accepted Answer

Matrix multiplication

Answer

Matrix subtraction

Answer

Matrix addition

Answer

Matrix division

Question 42

Which statement about groups is accurate?

Accepted Answer

Every group has a unique identity element

Answer

Every group is finite

Answer

Every group is abelian

Answer

Every group is cyclic