Groups and Rings: Master the Fundamentals of Algebraic Structures

Question 1

Which of the following is NOT a type of Lie algebra?

Accepted Answer

Commutative Lie algebra

Answer

Simple Lie algebra

Answer

Semisimple Lie algebra

Answer

Nilpotent Lie algebra

Question 2

Consider the ring Z_6. What is the additive inverse of 3?

Accepted Answer

5

Answer

0

Answer

3

Answer

4

Question 3

Which set serves as a minimal generating set for the group Z_6?

Accepted Answer

{1}

Answer

{1, 2}

Answer

{1, 2, 3}

Answer

{1, 2, 4}

Question 4

Which of the following properties does NOT belong to a group?

Accepted Answer

Commutativity of the operation

Answer

Associativity of the operation

Answer

Existence of an identity element

Answer

Existence of an inverse element for every element

Question 5

Which property is NOT characteristic of a group?

Accepted Answer

Commutativity of the operation

Answer

Existence of an inverse element for each element

Answer

Existence of an identity element

Answer

Associativity of the operation

Question 6

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

Accepted Answer

The ring of even integers

Answer

The ring of polynomials over the rational numbers

Answer

The ring of rational numbers

Answer

The ring of complex numbers

Question 7

Which of the following is a homomorphism between the groups Z and Z_2?

Accepted Answer

f(x) = x mod 2

Answer

f(x) = 2x

Answer

f(x) = x + 1

Answer

f(x) = x^2

Question 8

Which of the following is a linear transformation?

Accepted Answer

T(x) = Ax, where A is a matrix

Answer

T(x) = |x|

Answer

T(x) = x^2

Answer

T(x) = sin(x)

Question 9

Which of the following properties is NOT essential for a set to be considered a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Answer

Closure under the operation

Question 10

Under addition, the set of integers, Z, forms what type of algebraic structure?

Accepted Answer

Abelian group

Answer

Non-Abelian group

Answer

Ring

Answer

Field

Question 11

In the ring of 2x2 matrices over the real numbers, which matrix serves as the identity element?

Accepted Answer

The 2x2 identity matrix

Answer

The matrix with all entries equal to 2

Answer

The 2x2 zero matrix

Answer

The diagonal matrix with 1s on the diagonal

Question 12

Which of the following sets is a subring of the ring of integers, Z?

Accepted Answer

The set of even integers

Answer

The set of prime numbers

Answer

The set of odd integers

Question 13

The polynomial ring, R[x], where R is a ring, is an example of what type of algebraic structure?

Accepted Answer

Ring

Answer

Vector space

Answer

Field

Answer

Group

Question 14

What is the standard notation for the group of permutations of n objects?

Accepted Answer

Sn

Answer

Zn

Answer

Rn

Answer

Pn

Question 15

Which of the following is a practical application of group theory?

Accepted Answer

Cryptography

Answer

Calculus

Answer

Differential equations

Answer

Linear algebra

Question 16

A defining characteristic of the field of complex numbers, C, is:

Accepted Answer

Being algebraically closed

Answer

Being a finite field

Answer

Having a characteristic of 2

Answer

Having a non-zero nilpotent element

Question 17

Which of the following is a property of a group?

Accepted Answer

Associativity

Answer

Existence of invertible elements

Answer

Commutativity

Answer

Existence of an identity element

Question 18

The set of integers under addition forms a group. Which of the following is the identity element of this group?

Accepted Answer

0

Answer

None of the above

Answer

1

Answer

-1

Question 19

Let G be a group with operation *. If a, b ∈ G, then the inverse of a * b is:

Accepted Answer

b⁻¹ * a⁻¹

Answer

a * b⁻¹

Answer

a⁻¹ * b⁻¹

Answer

b * a⁻¹

Question 20

Which of the following is a ring and a field?

Accepted Answer

The field of rational numbers (Q, +, *)

Answer

The field of real numbers (R, +, *)

Answer

The field of complex numbers (C, +, *)

Answer

The ring of integers (Z, +, *)

Question 21

Which of the following is a cyclic group of order 3?

Accepted Answer

(Z₃, +)

Answer

(Z₅, +)

Answer

(Z₄, +)

Answer

({1, -1}, *)

Question 22

A group homomorphism f: G → H maps the identity element of G to:

Accepted Answer

The identity element of H

Answer

The inverse of the identity element of H

Answer

Any element in H

Answer

None of the above

Question 23

The image of a group homomorphism f: G → H is:

Accepted Answer

The set of elements in H that are the image of at least one element in G

Answer

The set of all elements in H

Answer

The set of all elements in G

Question 24

A ring homomorphism f: R → S is a function that preserves both addition and multiplication. Which of the following conditions MUST f satisfy?

Accepted Answer

f(0_R) = 0_S and f(1_R) = 1_S

Answer

f(0_R) = 1_S and f(1_R) = 1_S

Answer

f(0_R) = 0_S and f(1_R) = 0_S

Answer

f(0_R) = 1_S and f(1_R) = 0_S