Group Operations: Binary Operations, Identity, and Inverse Elements

Question 1

In a group (G, *), if the inverse of element 'a' is 'b', then 'b * a' equals:

Accepted Answer

'e'

Answer

'a'

Answer

'b'

Answer

'a * b'

Question 2

Let (G, *) be a group and 'H' be a subset of G. If 'H' is closed under the operation * and contains the identity element of G, then 'H' is a:

Accepted Answer

Subgroup of G

Answer

Proper subgroup of G

Answer

Cyclic group

Answer

Abelian group

Question 3

The order of an element 'a' in a group (G, *) is the smallest positive integer 'n' such that:

Accepted Answer

'a^n = e'

Answer

'a^n = a'

Answer

'a^n = a^-1'

Answer

'a^n = 1'

Question 4

The center of a group (G, *) is the set of elements that commute with every element of G. Which of the following is NOT a property of the center of a group?

Accepted Answer

It is always a cyclic group

Answer

It is a subgroup of G

Answer

It is always an Abelian group

Answer

It is non-empty

Question 5

Which of the following is NOT a property of a binary operation defined on a set?

Accepted Answer

Invertibility

Answer

Associativity

Answer

Identity element

Answer

Closure

Question 6

Which of the following is NOT a necessary condition for a set with a binary operation to be a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Answer

Inverse element

Question 7

Let (G, *) be a group and a, b ∈ G. Which of the following statements is true?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 8

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

Accepted Answer

Set of positive real numbers (ℝ+)

Answer

Set of integers (ℤ)

Answer

Set of even integers (2ℤ)

Answer

Set of rational numbers (ℚ)

Question 9

Which of the following is not a property of a binary operation in the context of abstract algebra?

Accepted Answer

Injectivity

Answer

Closure

Answer

Associativity

Answer

Commutativity

Question 10

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

Accepted Answer

The set of non-zero real numbers under multiplication

Answer

The set of real numbers under addition

Answer

The set of complex numbers under addition

Answer

The set of integers modulo 3 under addition

Question 11

Which of the following is NOT a fundamental property of a group's binary operation?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Existence of identity element

Answer

Existence of inverse elements

Question 12

In a group (G, *), if the operation * is commutative (i.e., a * b = b * a for all a, b in G), which property holds?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Existence of identity element

Answer

Existence of inverse elements

Question 13

In a group (G, *), for elements a and b in G, if a * x = b and x * a = b, then which property holds?

Accepted Answer

Existence of inverse element

Answer

Associativity

Answer

Commutativity

Answer

Existence of identity element

Question 14

How does the associative property of a binary operation contribute to the formation of a group structure?

Accepted Answer

It allows the order of operations to be changed without altering the outcome.

Answer

It ensures that every element has a multiplicative inverse.

Answer

It identifies the unique identity element of the group.

Question 15

In cryptography, how are group operations used?

Accepted Answer

Creating secure encryption algorithms

Answer

Solving complex algebraic equations

Answer

Optimizing network setups

Question 16

Which of the following is NOT a property of binary operations in groups?

Accepted Answer

Commutativity

Answer

Closure

Answer

Associativity

Answer

Existence of identity element

Question 17

Consider the set of integers under addition. Which element serves as the identity element?

Accepted Answer

0

Answer

-1

Answer

No identity element exists

Answer

1

Question 18

Which of the following operations on the set of rational numbers IS NOT a group operation?

Accepted Answer

Subtraction

Answer

Multiplication

Answer

Addition

Answer

Division

Question 19

Which of the following sets forms a group under the given operation?

Accepted Answer

The set of complex numbers under multiplication

Answer

The set of even integers under subtraction

Answer

The set of real numbers under addition

Answer

The set of positive rational numbers under division

Question 20

In the group (Z, +), where Z represents the set of integers, which element is the identity element?

Accepted Answer

0

Answer

5

Answer

1

Answer

-1

Question 21

Consider the set of all invertible n x n matrices over the field R. Which of the following operations on this set is a binary operation?

Accepted Answer

Matrix multiplication

Answer

Subtraction

Answer

Scalar multiplication

Answer

Addition

Question 22

In the group (Z_6, ×), where Z_6 represents the integers modulo 6 and × denotes multiplication, the element 4 has which order?

Accepted Answer

3

Answer

2

Answer

6

Answer

4

Question 23

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

Accepted Answer

(Z_7, +)

Answer

(M_2(R), ×)

Answer

(R, +)

Answer

(Q, ×)

Question 24

Identify the binary operation on the set of real numbers.

Accepted Answer

Addition (+)

Answer

Integration

Answer

Differentiation

Answer

Exponentiation

Question 25

Which property guarantees the associativity of an operation?

Accepted Answer

(a ∘ b) ∘ c = a ∘ (b ∘ c)

Answer

a ∘ a = a

Answer

a ∘ b = b ∘ a

Answer

a ∘ (b ∘ c) = b ∘ (c ∘ a)

Question 26

In a group, the element that leaves any other element unchanged when combined is termed as:

Accepted Answer

Identity element

Answer

Commutative element

Answer

Conjugate element

Answer

Neutral element

Question 27

For any group element 'a', if there exists an element 'b' such that a ∘ b = b ∘ a = e (identity element), then 'b' is:

Accepted Answer

Inverse element of 'a'

Answer

Associative element of 'a'

Answer

Conjugate element of 'a'

Answer

Commutative element of 'a'

Question 28

Which set forms a group under multiplication?

Accepted Answer

Non-zero complex numbers

Answer

Positive integers

Answer

Rational numbers

Question 29

Which statement is NOT true for finite groups?

Accepted Answer

Every subgroup is cyclic

Answer

Every element has a finite order

Answer

The order of the group is finite

Answer

The identity element is unique

Question 30

Which of the following properties is NOT required for a binary operation * to be a group operation on a set G?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Existence of inverse elements

Answer

Existence of an identity element

Question 31

Let * be a binary operation on a set S. Which of the following conditions guarantees that * is associative?

Accepted Answer

a * (b * c) = (a * b) * c for all a, b, c in S

Answer

a * a = a for all a in S

Answer

a * b = b * a for all a, b in S

Question 32

Consider the set of integers Z with the operation of addition. What is the identity element?

Accepted Answer

0

Answer

There is no identity element

Answer

-1

Answer

1

Question 33

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

Accepted Answer

-5

Answer

5

Answer

1/5

Question 34

Let G be a group with operation * and let a and b be elements of G. Which of the following is always true?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 35

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

Accepted Answer

The set of non-zero rational numbers

Answer

The set of even integers

Answer

The set of integers

Answer

The set of natural numbers

Question 36

Let G be a group and let a, b be elements of G. If (a * b)^2 = a^2 * b^2, then which of the following is true?

Accepted Answer

a * b = b * a

Answer

a * b = e (identity element)

Answer

None of the above

Answer

a = b

Question 37

Which of the following statements is NOT true about group homomorphisms?

Accepted Answer

A group homomorphism must be surjective.

Answer

A group homomorphism maps inverses to inverses.

Answer

A group homomorphism preserves the group operation.

Answer

The identity element is mapped to the identity element under a group homomorphism.

Question 38

If a group G is generated by an element 'a' of order 6, what is the order of the element a^4?

Accepted Answer

3

Answer

2

Answer

6

Answer

4

Question 39

The set of all symmetries of a square forms a group under the operation of composition. What is the order of this group?

Accepted Answer

8

Answer

4

Answer

12

Answer

6

Question 40

If H and K are subgroups of a group G, then which of the following is always a subgroup of G?

Accepted Answer

H ∩ K

Answer

H - K

Answer

H ∪ K

Answer

{hk | h ∈ H, k ∈ K}

Question 41

Which of the following properties is required for a set G with a binary operation * to be considered a group?

Accepted Answer

Associativity

Answer

Identity element

Answer

Inverse element

Answer

Closure

Question 42

Let * be a binary operation on a set S. What is the definition of an identity element e in S with respect to *?

Accepted Answer

For all elements a in S, a * e = e * a = a.

Answer

For all elements a in S, a * e = 0.

Answer

For all elements a in S, a * e = e.

Answer

For all elements a in S, a * e = a * a.

Question 43

Consider the group (Z, +), where Z is the set of integers and + is addition. What is the inverse element of 5?

Accepted Answer

-5

Answer

0

Answer

5

Answer

10

Question 44

Let * be a binary operation on a set S. What is the definition of the inverse element a⁻¹ of an element a in S with respect to *?

Accepted Answer

a * a⁻¹ = a⁻¹ * a = e, where e is the identity element in S.

Answer

a * a⁻¹ = a

Answer

a * a⁻¹ = e * a

Question 45

Which of the following sets forms a group under addition modulo 5?

Accepted Answer

{0, 1, 2, 3, 4}

Answer

{0, 2, 4}

Answer

{1, 2, 3, 4}

Answer

{0, 1, 2, 3}

Question 46

The set of all non-zero real numbers under multiplication forms a group. What is the identity element of this group?

Accepted Answer

1

Answer

0

Answer

-1

Answer

∞

Question 47

In the group of integers modulo 7 (Z₇, +₇), what is the inverse element of 3?

Accepted Answer

4

Answer

5

Answer

1

Answer

6

Question 48

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

Accepted Answer

The group of invertible matrices of order 2 under matrix multiplication

Answer

The group of rational numbers under addition

Answer

The group of integers under addition

Question 49

Let * be a binary operation on a set S. Which of the following conditions MUST be true for * to be associative?

Accepted Answer

(a * b) * c = a * (b * c) for all a, b, c in S

Answer

a * a⁻¹ = a⁻¹ * a = e for all a in S

Answer

a * b = b * a for all a, b in S

Answer

a * e = e * a = a for all a in S

Question 50

What is the order of the group (Z₅, +₅) ?

Accepted Answer

5

Answer

4

Answer

6

Answer

∞