Groups and Rings: Master the Fundamentals of Algebraic Structures

Question 1

Which of the following is a homomorphism between groups G and H?

Accepted Answer

A function f: G -> H that preserves the group operation

Answer

A function f: G -> H that is bijective

Answer

A function f: G -> H that is injective

Answer

A function f: G -> H that is surjective

Question 2

Let G be a finite group of order n. Which statement about the relationship between G and the cyclic group Z_n is valid?

Accepted Answer

G may or may not be isomorphic to Z_n.

Answer

G is necessarily isomorphic to Z_n.

Answer

G is always a subgroup of Z_n.

Answer

Z_n is always a subgroup of G.

Question 3

In a ring R with unity, what can be said about its invertible elements?

Accepted Answer

They form a group under multiplication.

Answer

They form a ring under multiplication.

Answer

They form a vector space over the field of rational numbers.

Answer

They form a field.

Question 4

In a group G, the elements a and b are related by which condition if 'ab = ba'?

Accepted Answer

a and b commute.

Answer

a and b generate a commutative subgroup.

Answer

G is a commutative group.

Answer

a and b are in the center of G.

Question 5

In a ring R with unity, what can be said about its ideals?

Accepted Answer

They form a lattice.

Answer

They form a semilattice.

Answer

They form a Boolean algebra.

Answer

They form a ring.

Question 6

What is the defining property of an abelian group?

Accepted Answer

Commutativity of multiplication

Answer

Closure under multiplication

Answer

Inverses exist for all elements

Answer

Associativity of multiplication

Question 7

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

Accepted Answer

The symmetric group on three elements

Answer

The set of integers under addition

Answer

The set of complex numbers under multiplication

Answer

The set of real numbers under addition

Question 8

Determine the order of the group Z/6Z.

Accepted Answer

6

Answer

2

Answer

3

Answer

12

Question 9

What is the characteristic of the ring Z/pZ, where p is a prime number?

Accepted Answer

p

Answer

0

Answer

1

Answer

p^2

Question 10

Which of the following is **not** a property of a group?

Accepted Answer

Commutativity of the group operation

Answer

Closure under the group operation

Answer

Associativity of the group operation

Answer

Existence of an identity element

Question 11

What is the annihilator of the subgroup <a> of a group G?

Accepted Answer

{x in G | xa = e}

Answer

{x in G | ax = e}

Answer

{x in G | xa = ax}

Answer

{x in G | xa = a}

Question 12

What is the radical of the ring Z_12?

Accepted Answer

{0, 2, 4, 6, 8, 10}

Answer

0

Answer

6

Answer

12

Question 13

What is the quotient ring of Z_12 by the ideal <4>?

Accepted Answer

Z_3

Answer

Z_4

Answer

Z_6

Answer

Z_12

Question 14

Which of the following statements accurately describes the Chinese Remainder Theorem?

Accepted Answer

It guarantees a unique solution when the moduli are pairwise co-prime.

Answer

It can solve any system of linear congruences, regardless of the moduli values.

Answer

It is limited to systems of two linear congruences.

Answer

It can solve systems of non-linear congruences.

Question 15

Which of the following functions is an isomorphism between the groups (Z, +) and (2Z, +)?

Accepted Answer

f(x) = 2x

Answer

f(x) = x/2

Answer

f(x) = x^2

Answer

f(x) = x + 1

Question 16

Which of the following is **not** a defining property of groups?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Question 17

What is the order of the group Z_6?

Accepted Answer

6

Answer

12

Answer

3

Answer

2

Question 18

What is the characteristic of the ring Z_p, where p is a prime number?

Accepted Answer

p

Answer

2p

Answer

p+1

Answer

p-1

Question 19

Which of the following is a cyclic group?

Accepted Answer

Z_n, where n is a positive integer

Answer

The symmetric group on 3 elements (S_3)

Answer

The quaternion group (Q_8)

Answer

The dihedral group of order 4 (D_4)