Definition of a Group: Interactive Learning Quizzes for Advanced Mathematics

Question 1

Which of the following properties is NOT true for a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Existence of an identity element

Answer

Closure

Question 2

For a set (G, *) to be considered a group, which conditions must be satisfied?

Accepted Answer

Closure, Associativity, Existence of an identity element, Existence of inverse elements

Answer

Closure, Associativity, Existence of an identity element

Answer

Closure, Commutativity, Existence of an identity element

Question 3

Which of the following is an illustration of a group?

Accepted Answer

Set of integers under the operation of addition

Answer

Set of odd numbers under the operation of subtraction

Answer

Set of real numbers under the operation of division

Answer

Set of rational numbers under the operation of multiplication

Question 4

In a group (G, *), given elements a and b, what is the inverse of (a * b)?

Accepted Answer

(a * b)^-1 = b^-1 * a^-1

Answer

(a * b)^-1 = a^-1 * b

Answer

(a * b)^-1 = b * a

Answer

(a * b)^-1 = a^-1 * b^-1

Question 5

Consider the set of permutations of the set {1, 2, 3} under the composition of functions. Does this set form a group?

Accepted Answer

Yes

Answer

No, due to the absence of associativity

Answer

No, as the set lacks inverse elements

Answer

No, as the set does not have an identity element

Question 6

Which of the following is NOT a necessary condition for a group to be cyclic?

Accepted Answer

The group is commutative

Answer

Every element has finite order

Answer

The group possesses an identity element

Answer

The group can be generated by a single element

Question 7

In a group (G, *), if a * b = b * a for all a and b in G, what can be concluded about the group?

Accepted Answer

(G, *) is an abelian group

Answer

(G, *) is a cyclic group

Answer

a = b

Answer

(G, *) is a non-abelian group

Question 8

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

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Answer

Closure

Question 9

A cyclic group is a group generated by a single element. Which of the following sets forms a cyclic group under the operation of multiplication modulo 5?

Accepted Answer

{1, 2, 3, 4}

Answer

{0, 2, 4, 6}

Answer

{1, 3, 5, 7}

Answer

{0, 1, 2, 3}

Question 10

Lagrange's theorem states that the order of a subgroup divides the order of the group. What is a consequence of Lagrange's theorem?

Accepted Answer

The order of a subgroup divides the order of the group

Answer

The group is cyclic

Answer

The group is Abelian

Question 11

Which of the following accurately defines a group?

Accepted Answer

A group is a non-empty set equipped with an operation that satisfies closure, associativity, identity element, and inverse element properties.

Answer

A group is a collection of numbers that can be added together.

Answer

A group is a set of elements that can be combined using a specific operation.

Question 12

Which of the listed properties is NOT characteristic of a group?

Accepted Answer

Commutativity

Answer

Existence of an inverse element

Answer

Associativity

Answer

Existence of an identity element

Question 13

Consider the set {1, 2, 3} and the operation ∘ defined as ∘(a, b) = a + b - 1. Does this constitute a group under this operation?

Accepted Answer

No

Answer

Yes

Answer

Only if 0 is added to the set.

Answer

Only if the operation is changed to multiplication.

Question 14

Which operation exhibits the property of associativity?

Accepted Answer

Multiplication

Answer

Exponentiation

Answer

Subtraction

Answer

Division

Question 15

Which element serves as the identity element for the operation of addition?

Accepted Answer

0

Answer

2

Answer

-1

Answer

1

Question 16

Within the operation of addition, what is the inverse element of 3?

Accepted Answer

-3

Answer

1

Answer

0

Question 17

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

Accepted Answer

(2Z, +)

Answer

(Q, +)

Answer

(R, +)

Answer

(Z*, +)

Question 18

Which of the listed groups is a cyclic group?

Accepted Answer

The group of integers under addition modulo 5

Answer

The group of real numbers under addition

Answer

The group of complex numbers under multiplication

Question 19

Which group is considered abelian?

Accepted Answer

The group of integers under addition

Answer

The group of real numbers under multiplication

Answer

The group of rational numbers under subtraction

Answer

The group of complex numbers under addition

Question 20

In the context of a group G, the inverse of the identity element e is:

Accepted Answer

e

Answer

The outcome of applying the operation twice to e

Answer

The element equal to its own inverse

Answer

The element paired with e during the operation

Question 21

Identify an essential property of an associative operation in a set G:

Accepted Answer

For any elements a, b, and c in G, the order of operations does not affect the outcome: (a * b) * c = a * (b * c)

Answer

For any element a in G, a * a = a

Answer

For any elements a and b in G, a * b = b * a

Question 22

Which consequence directly stems from the closure property in a group?

Accepted Answer

For any elements a and b in G, the result of their operation, denoted as a * b, is also an element of G

Answer

For any element a in G, its inverse, denoted as a^-1, is also in G

Answer

The set G contains the inverse element for every element under the operation

Question 23

Given a group G and elements a and b within it, if a * b = b * a, what can be concluded?

Accepted Answer

G is considered an abelian group

Answer

G is considered a non-abelian group

Answer

Elements a and b are equal, i.e., a = b

Answer

The inverses of a and b are equal, i.e., a^-1 = b^-1

Question 24

Identify an instance of a group from the following:

Accepted Answer

The set of integers under the operation of addition

Answer

The set of real numbers under the operation of division

Answer

The set of rational numbers under the operation of subtraction

Answer

The set of even numbers under the operation of multiplication

Question 25

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

Accepted Answer

Commutativity

Answer

Associativity

Answer

Closure under the operation

Answer

Existence of an identity element

Question 26

Which of the following is an example of a group?

Accepted Answer

The set of integers under addition

Answer

The set of real numbers under subtraction

Answer

The set of rational numbers under multiplication

Question 27

What is the order of the group (Z_6, +_6)?

Accepted Answer

6

Answer

10

Answer

4

Answer

8

Question 28

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

Accepted Answer

The group of symmetries of a square

Answer

The group of rational numbers under multiplication

Answer

The group of integers under addition

Answer

The group of complex numbers under addition

Question 29

What is the formal definition of a group?

Accepted Answer

A non-empty set G together with an operation * on G that combines any two elements of G to produce a third element of G, and that satisfies the following properties:

1. Closure: For all a, b ∈ G, a * b ∈ G.
2. Associativity: For all a, b, c ∈ G, (a * b) * c = a * (b * c).
3. Identity element: There exists an element e ∈ G such that for all a ∈ G, a * e = e * a = a.
4. Inverse element: For each a ∈ G, there exists an element b ∈ G such that a * b = b * a = e, where e is the identity element.

Answer

A collection of objects that can be combined using a specific operation.

Answer

A set of elements that are related to each other by a certain rule.

Question 30

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

Accepted Answer

Commutativity

Answer

Closure under the operation

Answer

Existence of an identity element

Answer

Associativity

Question 31

What is the identity element of the group of rational numbers under multiplication?

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 32

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

Accepted Answer

The set of all even integers

Answer

The set of odd integers

Answer

The set of negative integers

Answer

The set of prime numbers

Question 33

Which of the following is a famous unsolved problem in the field of group theory?

Accepted Answer

The classification of finite simple groups

Answer

The Goldbach conjecture

Answer

Fermat's last theorem

Answer

The Riemann hypothesis

Question 34

Which property is NOT essential for a set to be considered a group under a binary operation *?

Accepted Answer

Commutativity

Answer

Associativity of the operation

Answer

Existence of an identity element

Answer

Closure under the operation

Question 35

A group G with a binary operation * is called Abelian if, for every element a and b in G:

Accepted Answer

a * b = b * a

Answer

a * b = e, where e is the identity element

Answer

a * b = a

Answer

a * b ≠ b * a

Question 36

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

Accepted Answer

0

Answer

-1

Answer

None of the above

Answer

1

Question 37

In the group of integers under addition, what is the inverse of the element 5?

Accepted Answer

-5

Answer

1/5

Answer

5

Question 38

Consider the set {1, 2, 3, 4} with the operation of multiplication modulo 5. Which element is NOT part of the resulting group?

Accepted Answer

0

Answer

2

Answer

1

Question 39

In a group G with a binary operation *, which statement is always true?

Accepted Answer

The identity element is unique.

Answer

The group has a finite number of elements.

Answer

Every element has exactly two inverses.

Answer

The group is commutative.

Question 40

Consider the set of all 2x2 matrices with non-zero determinants under matrix multiplication. This set forms a group. What is the identity element of this group?

Accepted Answer

[[1, 0], [0, 1]]

Answer

[[0, 0], [0, 0]]

Answer

[[0, 1], [1, 0]]

Question 41

If (a * b) * c = a * (b * c) holds true for all elements a, b, and c in a group G with a binary operation *, then the group is considered:

Accepted Answer

Associative

Answer

Cyclic

Answer

Finite

Answer

Commutative

Question 42

What defines a group in mathematics?

Accepted Answer

A non-empty set with an operation that satisfies closure, associativity, identity element, and inverse element properties.

Answer

A set with an operation that satisfies closure and associativity properties.

Answer

A set with an operation that satisfies commutativity and identity element properties.

Answer

A non-empty set with an operation that satisfies closure, associativity, and identity element properties.

Question 43

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

Accepted Answer

Closure

Answer

Associativity

Answer

Distributivity

Answer

Commutativity

Question 44

Which property is NOT necessarily true for all groups?

Accepted Answer

Commutativity

Answer

Closure

Answer

Identity element

Answer

Associativity

Question 45

Which operation CANNOT be used to define a group?

Accepted Answer

Subtraction

Answer

Addition

Answer

Multiplication

Answer

Division

Question 46

Which of these sets with an operation forms a group?

Accepted Answer

The set of integers with addition.

Answer

The set of real numbers with division.

Answer

The set of rational numbers with subtraction.

Answer

The set of complex numbers with multiplication.

Question 47

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

Accepted Answer

0

Answer

-1

Answer

1

Answer

2

Question 48

What is the inverse element of 5 in the group of integers under addition?

Accepted Answer

-5

Answer

1

Answer

0

Answer

10

Question 49

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

Accepted Answer

The set of even integers

Answer

The set of odd integers

Answer

The set of positive integers

Answer

The set of prime numbers

Question 50

Which statement about groups is always true?

Accepted Answer

Every group has a unique identity element.

Answer

Every group is commutative.

Answer

Every group has a finite number of elements.

Question 51

In which field does group theory have practical applications?

Accepted Answer

Cryptography

Answer

Calculus

Answer

Statistics

Answer

Linear algebra