Group Theory Quizzes: Test Your Understanding

Question 1

Which of the following homomorphisms from Z (the integers under addition) to Z2 (the integers modulo 2) has the kernel {..., -4, -2, 0, 2, 4, ...}?

Accepted Answer

f(x) = x mod 2

Answer

f(x) = |x| mod 2

Answer

f(x) = x^2 mod 2

Answer

f(x) = -x mod 2

Question 2

Which of the following is the correct definition of a group?

Accepted Answer

A collection of elements with a binary operation that is associative, has an identity element, and every element has an inverse.

Answer

A set of permutations that form a closed operation under composition.

Answer

A collection of elements that can be combined in any order to produce a unique result.

Answer

A set of integers that are closed under addition and multiplication.

Question 3

Let H and K be subgroups of a group G. Which of the following statements is true?

Accepted Answer

Their intersection is always a subgroup of G.

Answer

Their union is always a subgroup of G.

Answer

Their direct product is always a subgroup of G.

Answer

None of the above are always true.

Question 4

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

Accepted Answer

Commutativity

Answer

Closure

Answer

Associativity

Answer

Identity element

Question 5

Let H be a subgroup of a group G. Which of the following is a necessary and sufficient condition for H to be a normal subgroup of G?

Accepted Answer

The left cosets of H in G are also right cosets.

Answer

H is closed under the operation of G.

Answer

H is the only subgroup of G.

Answer

The order of H divides the order of G.

Question 6

Consider the group S3 of permutations on 3 elements. How many elements does S3 have?

Accepted Answer

6

Answer

3

Answer

9

Answer

12

Question 7

Which of the following is a theorem that relates the order of a group and the order of its subgroups?

Accepted Answer

Lagrange's theorem

Answer

Cayley's theorem

Answer

Sylow's theorem

Answer

Burnside's theorem

Question 8

Which of the following is a fundamental property of groups, but NOT a defining one?

Accepted Answer

Commutativity

Answer

Closure

Answer

Associativity

Answer

Identity element

Question 9

If G is a finite group of order p^n, where p is a prime number, what is the maximum possible order of the center of G?

Accepted Answer

p^(n-1)

Answer

p^n

Answer

p

Answer

1

Question 10

Which of the following is a necessary and sufficient condition for a group to be abelian?

Accepted Answer

Every element commutes with every other element.

Answer

Every subgroup is normal.

Answer

The group has no non-trivial subgroups.

Answer

The group has a unique identity element.

Question 11

Which of the following statements about the order of an element in a group is true?

Accepted Answer

It is always a factor of the order of the group.

Answer

It is always a multiple of the order of the group.

Answer

It is always equal to the number of elements in the group.

Answer

It is always greater than the order of the group.

Question 12

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

Accepted Answer

Commutativity of the operation

Answer

Closure under the group operation

Answer

Associativity of the operation

Answer

Inverse element for every element

Question 13

Which function is a homomorphism from the group (Z, +) to the group (R, +)?

Accepted Answer

f(x) = 2x

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = |x|

Question 14

The order of an element a in a group G is defined as:

Accepted Answer

The least positive integer n such that a^n = e, where e is the identity element.

Answer

The number of distinct elements in the cyclic subgroup generated by a.

Answer

The number of elements in the conjugacy class of a.

Answer

The number of subgroups of G containing a.

Question 15

The center of a group G is:

Accepted Answer

The set of all elements that commute with every element of G.

Answer

The set of all elements that are their own inverses.

Answer

The set of all elements of finite order.

Answer

The set of all subgroups of G.

Question 16

Which of the following is not a Sylow subgroup of the symmetric group S_5?

Accepted Answer

The alternating group A_5

Answer

The group of all 5-cycles.

Answer

The group of all 3-cycles.

Answer

The group of all 2-cycles.

Question 17

The order of the symmetric group S_3 is:

Accepted Answer

6

Answer

2

Answer

3

Answer

9

Question 18

Given group G and its normal subgroup H, which is a coset of H in G?

Accepted Answer

aH = {h ∈ G | h = a⁻¹ha for some h in H} for all a in G

Answer

aH = {a} for all a in G

Answer

aH ∩ bH = ∅ for all a, b in G

Answer

aH ∪ bH = G for all a, b in G

Question 19

Consider the group G with presentation <a, b | a² = e, b² = e, ab = ba>. What can be deduced from this presentation?

Accepted Answer

G is isomorphic to the Klein four group

Answer

G is a cyclic group of order 2

Answer

G is abelian

Answer

G is infinite

Question 20

Which of the following is a subgroup of the group of integers under addition (+)?

Accepted Answer

The set of even integers

Answer

The set of prime numbers

Answer

The set of rational numbers

Question 21

If H is a subgroup of a group G, which of the following is always true?

Accepted Answer

The identity element of G is also the identity element of H

Answer

Every element of G has an inverse in H

Answer

The order of H is equal to the order of G

Question 22

Which property is NOT characteristic of a group?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Closure

Answer

Existence of an identity element

Question 23

In a cyclic group, the order refers to:

Accepted Answer

The smallest positive integer n such that every element raised to the power of n equals the identity element.

Answer

The number of elements in the group.

Answer

The largest positive integer n such that every element raised to the power of n equals the identity element.

Answer

The number of generators in the group.

Question 24

For a group (G, *) and a subset H of G, which condition guarantees that H is a subgroup of G?

Accepted Answer

Both (A) and (C)

Answer

H is closed under the operation *

Answer

H contains the identity element of G

Answer

For every element a in H, its inverse a^(-1) is also in H

Question 25

The permutation group S_n represents:

Accepted Answer

The group of all permutations of the set {1, 2, ..., n}.

Answer

The group of all even permutations of the set {1, 2, ..., n}.

Answer

The group of all odd permutations of the set {1, 2, ..., n}.

Answer

The group of all subgroups of the set {1, 2, ..., n}.

Question 26

Determine the function that defines a homomorphism between the additive group of integers under addition and the multiplicative group of positive real numbers under multiplication:

Accepted Answer

f(x) = e^x

Answer

f(x) = 2^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 27

What is the definition of the centre of a group G?

Accepted Answer

The set of all elements of G that commute with every other element of G.

Answer

The set of all elements of G that are self-inverse.

Answer

The set of all subgroups of G.

Answer

The set of all homomorphisms from G to itself.

Question 28

Which type of subgroup is a subgroup of prime order?

Accepted Answer

A Sylow subgroup

Answer

A subgroup of maximal order

Answer

A subgroup that is normal in the group

Answer

A subgroup that is cyclic

Question 29

What type of group is the group of symmetries of an equilateral triangle?

Accepted Answer

Dihedral of order 6

Answer

Cyclic of order 3

Answer

Symmetric of order 3

Answer

Alternating of order 3

Question 30

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

Accepted Answer

Commutativity of the binary operation

Answer

Associativity of the binary operation

Answer

Existence of an identity element

Answer

Existence of an inverse element for each element

Question 31

Let (G, *) be a group. Which of the following is a defining property of a subgroup of G?

Accepted Answer

It is a non-empty subset of G that is closed under * and contains the identity element of G

Answer

It is a non-empty subset of G that is closed under *

Answer

It is a non-empty subset of G that contains the identity element of G

Answer

It is any non-empty subset of G

Question 32

Let (G, *) and (H, #) be groups. Which of the following defines a homomorphism from G to H?

Accepted Answer

A function f: G -> H such that f(a * b) = f(a) # f(b) for all a, b in G

Answer

A function f: G -> H such that f(a * b) = f(b) * f(a) for all a, b in G

Answer

A function f: G -> H such that f(a^(-1)) = f(a)^(-1) for all a in G

Answer

Any function f: G -> H

Question 33

Which of the following is a key characteristic of a cyclic group?

Accepted Answer

It can be generated by a single element

Answer

Its order is prime

Answer

It is isomorphic to a group of integers

Answer

It is always a finite group

Question 34

Determine the order of the cyclic group generated by the permutation (1 2 3 4 5).

Accepted Answer

5

Answer

3

Answer

4

Answer

6

Question 35

Let G be a group and H a subgroup of G. What is a subset of G of the form gH for some g in G called?

Accepted Answer

Coset of H in G

Answer

Subgroup of H

Answer

Complement of H in G

Answer

Non-empty subset of G

Question 36

Identify a necessary and sufficient condition for a finite group G of order n to be cyclic.

Accepted Answer

G contains an element of order n

Answer

G is abelian

Answer

G is simple

Answer

G is isomorphic to a subgroup of the multiplicative group of complex numbers of modulus 1

Question 37

Let H = {e, (1 2)} and K = {e, (1 3)} be subgroups of the group G = {e, (1 2), (1 3), (2 3), (1 2 3)}. Which of the following is a homomorphism from H to K?

Accepted Answer

h(e) = e, h((1 2)) = (1 3)

Answer

h(e) = e, h((1 2)) = (1 2)

Answer

h(e) = (1 2), h((1 2)) = e

Answer

h(e) = (1 3), h((1 2)) = (1 2)

Question 38

The order of a cyclic group generated by an element a is defined as:

Accepted Answer

The smallest positive integer n such that a^n = e

Answer

The smallest positive integer n such that a^n = a

Answer

The number of elements in the group

Answer

Always infinity

Question 39

One consequence of Lagrange's theorem is that:

Accepted Answer

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

Answer

The order of an element divides the order of the group.

Answer

The group has a unique subgroup of prime order.

Answer

The group has a normal subgroup of index 2.

Question 40

Which of the given functions is a homomorphism from the group (Z, +) to the group (R+, ×)?

Accepted Answer

f(x) = x

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 41

Let G be a group and f: G -> H be a function. Which of the following conditions is necessary for f to be a group homomorphism?

Accepted Answer

f(a * b) = f(a) * f(b) for all a, b in G

Answer

f(e) = e, where e is the identity element of G

Answer

f^-1(e) = e, where e is the identity element of H

Answer

f is injective

Question 42

Which of the following sets forms a permutation group of order 3?

Accepted Answer

S_3

Answer

Z_3

Answer

D_3

Answer

Q_8

Question 43

Which of the following is an invariant subgroup of S_5?

Accepted Answer

A_5

Answer

Z_5

Answer

D_5

Answer

Q_8

Question 44

Which of the following is a free group of rank 2?

Accepted Answer

F_2

Answer

Z_2 * Z_2

Answer

S_3

Answer

D_4

Question 45

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

Accepted Answer

Commutativity

Answer

Associativity

Answer

Identity element

Answer

Closure

Question 46

If G is a group and H is a normal subgroup of G, then the set of all right cosets of H in G forms:

Accepted Answer

A partition of G

Answer

A group

Answer

A subgroup of G

Answer

A normal subgroup of G

Question 47

Consider the set {a, b, c} with the operation ∘ defined as follows: a ∘ b = b, b ∘ c = a, c ∘ a = b. Which of the following accurately describes this set with respect to group theory?

Accepted Answer

It forms an abelian group.

Answer

It forms a cyclic group.

Answer

It does not form a group.

Answer

It forms a permutation group.

Question 48

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

Accepted Answer

The group of symmetries of a regular hexagon

Answer

The group of integers under addition

Answer

The group of all real numbers under multiplication

Answer

The group of even integers under addition