Master Group Properties: Enhance Your Algebraic Structures Knowledge

Question 1

Precise: Which fundamental property characterizes all groups?

Accepted Answer

Unaffected: Associativity

Answer

Reworded: Idempotence

Answer

Reworded: Distributivity

Answer

Reworded: Commutativity

Question 2

Identify the identity element of the group (Z, +).

Accepted Answer

Unaffected: 0

Answer

Reworded: -1

Answer

Removed: 2

Answer

Unaffected: 1

Question 3

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

Accepted Answer

Unaffected: 5

Answer

Reworded: -10

Answer

Removed: 0

Answer

Reworded: 10

Question 4

Find the order of the group (Z_6, +).

Accepted Answer

Unaffected: 6

Answer

Removed: 18

Answer

Removed: 3

Answer

Removed: 12

Question 5

Precise: Which group among the options is a cyclic group of order 5?

Accepted Answer

Unaffected: (Z_5, +)

Answer

Reworded: (Z_8, +)

Answer

Removed: (Z_6, +)

Answer

Reworded: (Z_7, +)

Question 6

Precise: Is the group (D_6, ∘) abelian?

Accepted Answer

Unaffected: No

Answer

Reworded: No, if the identity element is included

Answer

Reworded: Yes

Answer

Reworded: Yes, if the identity element is removed

Question 7

How does the associative property of groups affect the grouping of elements?

Accepted Answer

The order of grouping elements is irrelevant due to the associative property.

Answer

The associative property applies only to certain grouping arrangements of elements.

Answer

The order of grouping elements can alter the result of the operation.

Question 8

Which of the following is a fundamental property of groups that defines how the group operation interacts with itself?

Accepted Answer

Associativity

Answer

Distributivity

Answer

Idempotency

Answer

Commutativity

Question 9

In a group (G, *), for any elements a, b, c in G, which equation represents the associative property?

Accepted Answer

(a * b) * c = a * (b * c)

Answer

a * b = b * a

Answer

a * (b * c) = c * (a * b)

Answer

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

Question 10

If a group G consists of n elements, what is the maximum number of distinct subgroups that can exist in G?

Accepted Answer

2^n

Answer

n!

Answer

n^2

Answer

n + 1

Question 11

Which property do all abelian groups exhibit?

Accepted Answer

Commutativity

Answer

Inverse element

Answer

Identity element

Answer

Associativity

Question 12

Calculate the order of the finite group (Z_6, +_6), where Z_6 is the set of integers modulo 6 under the operation of addition modulo 6.

Accepted Answer

6

Answer

12

Answer

2

Answer

3

Question 13

Which property is NOT characteristic of groups?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Closure under the group operation

Answer

Existence of an identity element

Question 14

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

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 15

Which condition is necessary and sufficient for a group (G, *) to be abelian?

Accepted Answer

For all a, b in G, a * b = b * a

Answer

For all a, b, c in G, (a * b) * c = a * (b * c)

Answer

There exists an element e in G such that for all a in G, a * e = a

Answer

For all a in G, a * a = a

Question 16

What is the order of the group (Z_8, +)?

Accepted Answer

8

Answer

16

Answer

2

Answer

4

Question 17

Which of the following is NOT a property of groups?

Accepted Answer

Commutativity

Answer

Inverse element

Answer

Associativity

Answer

Identity element

Question 18

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

Accepted Answer

The set of all non-zero real numbers, excluding 1

Answer

The set of all even integers

Answer

The set of all integers

Answer

The set of all prime numbers

Question 19

Which of the following is a consequence of the associative property in groups?

Accepted Answer

Parentheses can be omitted without changing the value of an expression.

Answer

The order of elements in a group does not matter.

Answer

Every group has a unique identity element.

Question 20

Let (G, *) be a group. Which of the following statements is equivalent to the statement that every element in G has an inverse element?

Accepted Answer

For every a in G, there exists an element b in G such that a * b = e and b * a = e.

Answer

For every a in G, there exists an element b in G such that a * b = b * a = e.

Answer

For every a in G, there exists an element b in G such that a * b = e.

Question 21

Which of the following is a property of cyclic groups?

Accepted Answer

Every element can be generated by repeated application of a single generator.

Answer

The order of the group is always prime.

Answer

The group is always commutative.

Question 22

Which of the following is an application of group theory in computer science?

Accepted Answer

Cryptography

Answer

Database management

Answer

Software engineering

Answer

Artificial Intelligence

Question 23

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

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Question 24

Consider the set G = {1, -1, i, -i} under the operation of multiplication. Which statement accurately describes G?

Accepted Answer

G is a group.

Answer

G is a group under the operation of addition, not multiplication.

Answer

G is associative but does not have an identity element.

Answer

G does not have inverse elements and thus is not a group.

Question 25

Which element serves as the identity element for the group (R+, ×)?

Accepted Answer

1

Answer

0

Answer

2

Answer

-1

Question 26

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

Accepted Answer

The group of positive integers co-prime with 10

Answer

The group of complex numbers.

Answer

The group of negative integers.

Answer

The group of rational numbers.

Question 27

Which of the following is NOT a fundamental property of groups?

Accepted Answer

Commutativity

Answer

Identity element

Answer

Associativity

Answer

Inverse element

Question 28

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

Accepted Answer

The group of integers under addition modulo 5

Answer

The group of complex numbers under addition

Answer

The group of rational numbers under multiplication

Answer

The group of real numbers under addition

Question 29

Let G be a group and let H be a subgroup of G. Which of the following statements about the orders of groups and subgroups is true?

Accepted Answer

The order of H divides the order of G.

Answer

The order of G is always equal to the order of H.

Answer

The order of H is always a multiple of the order of G.

Answer

The order of G is always a factor of the order of H.

Question 30

Which of the following is a consequence of Lagrange's Theorem?

Accepted Answer

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

Answer

The group of integers under addition has an infinite number of subgroups.

Answer

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

Answer

Any group of prime order is cyclic.

Question 31

Which of the following is an application of group theory?

Accepted Answer

Error-correcting codes

Answer

Calculating the area of a triangle

Answer

Solving differential equations

Answer

Finding the volume of a sphere

Question 32

Which of the following is a defining property of all groups?

Accepted Answer

Associativity

Answer

Idempotence

Answer

Distributivity

Answer

Commutativity

Question 33

Which of the following is NOT a property of groups?

Accepted Answer

Distributivity

Answer

Existence of identity element

Answer

Associativity

Answer

Closure under the operation

Question 34

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

Accepted Answer

0

Answer

1

Answer

2

Answer

-1

Question 35

Consider the set {a, b, c} with the operation defined as follows:
a * b = c
b * c = a
c * a = b
Does this set and operation form a group?

Accepted Answer

Yes, it is a group.

Answer

No, it is not a group because it is not closed under the operation.

Answer

No, it is not a group because it does not have an inverse element for every element.

Answer

No, it is not a group because it does not have an identity element.

Question 36

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

Accepted Answer

The group of 2x2 matrices with determinant 1 under matrix multiplication.

Answer

The group of complex numbers of modulus 1 under multiplication.

Answer

The group of rational numbers under addition.

Question 37

What is the order of the cyclic group generated by the element a, where a^4 = e ?

Accepted Answer

4

Answer

10

Answer

2

Answer

7

Question 38

Consider the group G = {1, -1, i, -i} under multiplication.

Accepted Answer

It is a cyclic group of order 4.

Answer

It is an abelian group of order 2.

Answer

It is a non-abelian group of order 4.

Question 39

Which of the following is an application of group theory?

Accepted Answer

Symmetry analysis in chemistry

Answer

Statistics

Answer

Calculus

Answer

Probability

Question 40

Which of the following properties is **not** required for a set to be considered a group under a given operation?

Accepted Answer

Commutativity

Answer

Inverse element

Answer

Associativity

Answer

Identity element

Question 41

In the group of integers under addition (ℤ, +), what is the identity element?

Accepted Answer

0

Answer

1

Answer

2

Answer

-1

Question 42

In the group of permutations of a set with 3 elements (S3, ∘), which permutation is its own inverse?

Accepted Answer

(1 2 3)

Answer

(1 2)

Answer

(1 3)

Question 43

Which of the following statements about groups is always true?

Accepted Answer

Every group has a unique identity element.

Answer

Every group is commutative (abelian).

Answer

Every group has a finite number of elements.

Answer

Every group can be generated by a single element (cyclic).

Question 44

Consider the group of positive rational numbers under multiplication (Q+, ·). Which of the following elements is **not** a member of this group?

Accepted Answer

-1

Answer

1

Answer

2

Answer

0

Question 45

Which of the following operations, when applied to the set {a, b, c, d}, would define a group?

Accepted Answer

a ∘ b = b ∘ a (commutative operation)

Answer

a ∘ b = b (not closed, no inverses for a, c, d)

Answer

a ∘ b = c (not closed, no identity)

Answer

a ∘ b = a (not closed, no inverses for b, c, d)

Question 46

In the group of integers modulo 5 (ℤ5, +), what is the inverse of the element 3?

Accepted Answer

2

Answer

4

Answer

0

Answer

3

Question 47

Which of the following groups is a cyclic group?

Accepted Answer

Group of integers under addition (ℤ, +)

Answer

Group of complex numbers of modulus 1 under multiplication (U, ·)

Answer

Group of real numbers under multiplication (ℝ, ·)

Answer

Group of permutations of a set with 3 elements (S3, ∘)

Question 48

Is the following statement true or false? 'Every subgroup of a group is also a group.'

Accepted Answer

True

Answer

True only if the original group is commutative (abelian)

Answer

False

Answer

True only if the original group is finite

Question 49

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

Accepted Answer

Group of 2 × 2 matrices with determinant 1 under matrix multiplication (SL2(ℝ), ·)

Answer

Group of permutations of a set with 2 elements (S2, ∘)

Answer

Group of complex numbers of modulus 1 under multiplication (U, ·)

Answer

Group of integers under addition (ℤ, +)

Question 50

Which of the following properties is NOT a defining characteristic of a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Inverse element

Answer

Identity element

Question 51

Let G be a group with operation *. If a, b, and c are elements of G, which of the following equations represents the associative property?

Accepted Answer

(a * b) * c = a * (b * c)

Answer

a * a⁻¹ = e

Answer

a * e = a

Answer

a * b = b * a

Question 52

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

Accepted Answer

-5

Answer

5

Answer

1

Answer

0

Question 53

Consider the group (Z₄, +₄), where +₄ represents addition modulo 4. Which of the following is the identity element of this group?

Accepted Answer

0

Answer

1

Answer

3

Answer

2

Question 54

Let G be a group with operation *. Which of the following statements is ALWAYS true?

Accepted Answer

Every element in G has a unique inverse.

Answer

The group operation is commutative.

Answer

The identity element of G is the only element that is its own inverse.

Answer

The group G is finite.

Question 55

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

Accepted Answer

The set of all non-zero real numbers with multiplication.

Answer

The set of all positive integers with division.

Answer

The set of all integers with subtraction.

Question 56

In a group G, if a² = e, where e is the identity element, then which of the following must be true?

Accepted Answer

a is its own inverse.

Answer

a is the inverse of another element in G.

Answer

a is the identity element.

Answer

The order of a is 2.

Question 57

Which of the following is NOT a subgroup of the group (Z, +)?

Accepted Answer

The set of all odd integers.

Answer

The set of all even integers.

Answer

The set of all multiples of 3.

Answer

The set containing only the element 0.

Question 58

What is the order of the element 3 in the group (Z₇, +₇), where +₇ represents addition modulo 7?

Accepted Answer

7

Answer

6

Answer

3

Answer

1

Question 59

The group (Z₅, +₅) is an example of which type of group?

Accepted Answer

Cyclic group

Answer

Dihedral group

Answer

Abelian group

Answer

Symmetric group