Algebra Quizzes: Test Your Understanding of Algebraic Structures

Question 1

What enzyme is responsible for unwinding and separating the DNA double helix during replication?

Accepted Answer

Helicase

Answer

Transcription

Answer

Ligase

Question 2

Which of the following is NOT an application of Fermat's Little Theorem?

Accepted Answer

Finding the greatest common divisor of two numbers

Answer

Primality testing

Answer

Finding primitive roots

Answer

Solving Diophantine equations

Question 3

Which of the following is NOT a fundamental algebraic structure covered in this unit?

Accepted Answer

Vector spaces

Answer

Groups

Answer

Rings

Answer

Fields

Question 4

Prove that the set of all invertible matrices under matrix multiplication satisfies the properties of a group.

Accepted Answer

Closure: The product of any two invertible matrices is invertible.

Accepted Answer

Identity element: The identity matrix is invertible and leaves any other matrix unchanged when multiplied.

Accepted Answer

Inverse: Every invertible matrix has an inverse matrix that is also invertible. Multiplying an invertible matrix by its inverse yields the identity matrix.

Question 5

Which algebraic concept involves a set of elements that can be combined in a specific way, and the result of the combination is also in the set?

Accepted Answer

Group

Answer

Field

Answer

Ring

Question 6

If every element in a group has an inverse, what must be true about the group operation?

Accepted Answer

The operation has an identity element.

Answer

The operation is associative.

Answer

The operation is commutative.

Answer

The group is cyclic.

Question 7

Determine the order of the cyclic group generated by the element a in the group (Z_6, +_6), where Z_6 = {0, 1, 2, 3, 4, 5} and +_6 denotes addition modulo 6.

Accepted Answer

3

Answer

2

Answer

4

Answer

6

Question 8

Which of the following is not a subfield of algebra?

Accepted Answer

Analysis

Answer

Linear algebra

Answer

Number theory

Answer

Algebraic topology

Question 9

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

Accepted Answer

It contains zero divisors.

Answer

It forms an abelian group under addition.

Answer

It is closed under multiplication.

Answer

It has a multiplicative identity.

Question 10

Define a group.

Accepted Answer

A set with an associative operation, an identity element, and inverse elements.

Answer

A set with an associative operation and an identity element.

Answer

A set with a commutative operation and an identity element.

Answer

A set with an associative operation and a zero element.

Question 11

In a group G with identity element e, which property holds true?

Accepted Answer

For all a in G, a * e = a and e * a = a

Answer

There exists an a in G such that a * a = e

Answer

For all a and b in G, (a * b) * e = a * b

Answer

There exists an a in G such that a * a = a

Question 12

Consider the group G = {0, 1, 2, 3, 4} under the operation of addition modulo 5. Is G cyclic? If so, find a generator for G.

Accepted Answer

G is cyclic with generator 2.

Answer

G is not cyclic.

Answer

G is cyclic with generator 3.

Answer

G is cyclic with generator 4.

Question 13

In the group (ℤ, +), what is the additive inverse of 5? Determine the element in the group that, when added to 5, results in the additive identity, 0.

Accepted Answer

-5

Answer

0

Answer

5

Answer

1

Question 14

In the ring (ℚ, +, ×), what is the multiplicative identity? Identify the element in the ring that, when multiplied by any other element, leaves that element unchanged.

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 15

Which of the following sets is a quotient ring of the ring (ℤ, +, ×)? Determine the set that results from factoring out an ideal in the ring, preserving the ring structure.

Accepted Answer

The set of rational numbers, (ℚ, +, ×)

Answer

The set of integers, (ℤ, +, ×)

Answer

The set of real numbers, (ℝ, +, ×)

Answer

The set of complex numbers, (ℂ, +, ×)

Question 16

What is the order of the group (ℤ₄, +)? Determine the number of elements in the group.

Accepted Answer

4

Answer

1

Answer

2

Answer

3

Question 17

Which of the following maps is an automorphism of the group (ℤ₂, +)? Identify the map that preserves the group structure, maintaining the group operation and identity element.

Accepted Answer

The map f(x) = -x

Answer

The map f(x) = x + 1

Answer

The map f(x) = x - 1

Answer

The map f(x) = 1 - x

Question 18

In a group, which property is NOT satisfied?

Accepted Answer

Commutativity of the operation

Answer

Associativity of the operation

Answer

Existence of an identity element

Answer

Existence of inverse elements

Question 19

In a ring, the operation of addition is known as:

Accepted Answer

Addition

Answer

Multiplication

Answer

Subtraction

Answer

Division

Question 20

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 set of complex numbers under addition

Answer

The set of rational numbers under multiplication

Question 21

Which of the following is a subring of the ring of integers?

Accepted Answer

The set of even integers

Answer

The set of odd integers

Answer

The set of positive integers

Answer

The set of negative integers

Question 22

Which of the following algebraic structures is a non-abelian group?

Accepted Answer

(S₃, ○)

Answer

(ℤ, +)

Answer

(ℤ₂, ×)

Answer

(ℝ, +)

Question 23

In a ring with a unity element, which of the following statements is always true?

Accepted Answer

The ring has no zero divisors.

Answer

The ring is a field.

Answer

The ring is commutative.

Answer

The ring is isomorphic to the integers.

Question 24

Identify the finite field from the following options:

Accepted Answer

(ℤ₂, +, ×)

Answer

(ℚ, +, ×)

Answer

(ℝ, +, ×)

Answer

(ℂ, +, ×)

Question 25

Which condition is equivalent to a group being cyclic?

Accepted Answer

The group is isomorphic to the integers under addition.

Answer

The group has a finite number of elements.

Answer

The group has an element of infinite order.

Answer

The group is commutative.

Question 26

Which property of a ring implies that it is a field?

Accepted Answer

Absence of zero divisors

Answer

Commutativity

Answer

Existence of a multiplicative identity

Answer

Vector space structure over ℚ

Question 27

Identify the prime ideal from the given options:

Accepted Answer

(2) in ℤ

Answer

(0) in ℤ

Answer

(x²) in ℝ[x]

Answer

(x, y) in ℝ[x, y]

Question 28

Which of the following algebraic structures is a subfield of a given field F?

Accepted Answer

ℚ

Answer

ℤ

Answer

ℝ

Answer

ℂ

Question 29

Which property is NOT applicable to groups?

Accepted Answer

Commutative property

Answer

Associative property

Answer

Identity element

Answer

Inverse element

Question 30

Determine the order of the group (Z/4Z, +).

Accepted Answer

4

Answer

2

Answer

6

Answer

8

Question 31

Find the eigenvalues of the matrix A = [2 1; -1 2].

Accepted Answer

1, 3

Answer

1, -3

Answer

3, 1

Answer

-3, 1

Question 32

Find the null space of the matrix A = [1 2; 3 4].

Accepted Answer

{(x, -2x)}

Answer

{(x, 2x)}

Answer

{(x, -x)}

Answer

{(x, x)}

Question 33

Find the quotient ring ℤ₂[x]/(x² + x + 1).

Accepted Answer

ℤ₂[x]/(x² + x + 1)

Answer

ℤ₂[x]/(x)

Answer

ℤ₂[x]/(x²)

Answer

ℤ₂[x]/(x² + x)

Question 34

Which group action maps every element of the set to the same element?

Accepted Answer

Trivial action

Answer

Faithful action

Answer

Effective action

Answer

Regular action

Question 35

Which of the following fields extends the field of rational numbers (ℚ)?

Accepted Answer

ℝ (set of real numbers)

Accepted Answer

ℂ (set of complex numbers)

Answer

ℤ (set of integers)

Answer

ℤ₄ (set of integers modulo 4)

Question 36

Which property is NOT required for a set to be considered a group?

Accepted Answer

Commutativity of the operation

Answer

Closure under the operation

Answer

Associativity of the operation

Answer

Existence of an identity element

Question 37

Which of the following groups exhibits both abelian and cyclic properties?

Accepted Answer

Integers under addition

Answer

Symmetric group with three elements

Answer

Complex numbers with modulus 1 under multiplication

Answer

Quaternions under multiplication

Question 38

Which property is NOT shared by all groups?

Accepted Answer

Commutativity

Answer

Closure

Answer

Associativity

Answer

Identity element

Question 39

In a group, is the inverse of an element unique?

Accepted Answer

Yes

Answer

No

Question 40

Which subset of the ring of integers is also a subring?

Accepted Answer

Set of even integers

Answer

Set of odd integers

Answer

Set of prime numbers

Answer

Set of rational numbers

Question 41

Which statement is a direct consequence of Lagrange's Theorem?

Accepted Answer

In a finite group, the order of each element divides the order of the group.

Answer

Every finite group is cyclic.

Answer

Every group with prime order is cyclic.

Answer

Every group of odd order is solvable.

Question 42

What is the centre of a group?

Accepted Answer

The set of elements commuting with every other element in the group.

Answer

The set of elements equal to their own inverses.

Answer

The set of elements with finite order.

Answer

The set of elements generating the group.

Question 43

Which subgroup is a normal subgroup?

Accepted Answer

Invariant under conjugation by every element of the group.

Answer

Having the same order as the group.

Answer

Contained in every subgroup of the group.

Answer

Generated by a single element.

Question 44

What is the definition of the quotient group G/N?

Accepted Answer

Set of all cosets of N in G.

Answer

Set of all subgroups of G containing N.

Answer

Set of all elements of G not in N.

Answer

Set of all automorphisms of G fixing N.

Question 45

Which set forms a cyclic group?

Accepted Answer

The set of complex numbers of the form e^(iθ) where θ is an arbitrary real number, under multiplication.

Answer

The set of rational numbers under addition.

Answer

The set of integers under multiplication.

Question 46

Which function is an isomorphism between the groups (Z, +) and (R, +)?

Accepted Answer

f(x) = x

Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Question 47

Which set is a quotient group of the group (Z, +)?

Accepted Answer

The set of integers modulo 3.

Answer

The set of matrices with real entries.

Answer

The set of complex numbers.

Answer

The set of rational numbers.

Question 48

Which set is a free group on two generators?

Accepted Answer

The set of all words in the alphabet {a, b} under concatenation, with the empty word as the identity element.

Answer

The set of all words in the alphabet {a, b} under concatenation, without the empty word.

Question 49

Which property is NOT applicable to a group?

Accepted Answer

Commutativity

Answer

Existence of inverse elements

Answer

Presence of an identity element

Answer

Associativity of the operation

Question 50

Which set qualifies as a field?

Accepted Answer

The set of rational numbers

Answer

The set of integers

Answer

The set of prime numbers

Answer

The set of even integers

Question 51

In the context of group theory, what is a crucial property of the order of an element?

Accepted Answer

It is a divisor of the order of the group

Answer

It is equal to the number of elements in the group

Answer

It is always a prime number

Answer

It is always even

Question 52

In group theory, a homomorphism is a function between groups that preserves the group operation. Which function represents a homomorphism from the additive group of integers to the multiplicative group of positive real numbers?

Accepted Answer

f(x) = e^x

Answer

f(x) = sin(x)

Answer

f(x) = x!

Answer

f(x) = x^2

Question 53

Which of the following is a commutative group?

Accepted Answer

The set of integers under addition

Answer

The set of real numbers under subtraction

Answer

The set of complex numbers under division

Answer

The set of rational numbers under multiplication

Question 54

Which of the following is a field?

Accepted Answer

The set of rational numbers

Answer

The set of integers

Answer

The set of irrational numbers

Answer

The set of imaginary numbers

Question 55

Which of the following is an example of an isomorphism?

Accepted Answer

A bijective homomorphism

Answer

An endomorphism

Answer

A surjective homomorphism

Answer

An injective homomorphism

Question 56

Which of the following is NOT a subring of the ring of integers?

Accepted Answer

The set of odd integers

Answer

The set of even integers

Answer

The set of positive integers

Question 57

Which of the following is a characteristic of a finite field?

Accepted Answer

It has a prime number of elements

Answer

It has an infinite number of elements

Answer

It has an even number of elements

Answer

It has an odd number of elements

Question 58

Which of the following is an example of a linear transformation?

Accepted Answer

A function that preserves vector addition and scalar multiplication

Answer

A function that preserves vector addition only

Answer

A function that preserves scalar multiplication only

Question 59

Which of the following is a property of an eigenvalue?

Accepted Answer

It is a solution to the characteristic equation of the associated matrix

Answer

It is always a negative number

Answer

It is always a positive number

Question 60

Which of the following is NOT a requirement for a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Inverse element

Answer

Identity element

Question 61

Which of the following is an example of a ring that is NOT a field?

Accepted Answer

The set of integers with standard addition and multiplication

Answer

The set of rational numbers with standard addition and multiplication

Answer

The set of complex numbers with standard addition and multiplication

Answer

The set of real numbers with standard addition and multiplication

Question 62

Which of the following functions represents a linear transformation?

Accepted Answer

f(x) = Ax, where A is a matrix and x is a vector

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Question 63

A matrix is considered diagonalizable if it has:

Accepted Answer

A complete set of linearly independent eigenvectors

Answer

All eigenvalues equal to zero

Answer

Only one distinct eigenvalue

Answer

Complex eigenvalues

Question 64

What defines a group homomorphism?

Accepted Answer

A function between two groups that preserves the group operation

Answer

A one-to-one function between two groups

Answer

An onto function between two groups

Answer

A function between two groups that maps the identity element of the first group to the identity element of the second group