Subgroups in Abstract Algebra: Definition, Examples, and Lagrange's Theorem

Question 1

Within the group (Z, +) under addition, which of these subsets is a subgroup?

Accepted Answer

The set of all even integers

Answer

The set of all prime integers

Answer

The set of all positive integers

Answer

The set of all negative integers

Question 2

Accurately state Lagrange's Theorem.

Accepted Answer

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

Answer

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

Answer

The order of a subgroup equals the order of the group

Answer

The order of a group equals the product of the orders of its subgroups

Question 3

Which of the following subsets is not a subgroup of (Q*, ⋅), where Q* represents the group of non-zero rational numbers under multiplication?

Accepted Answer

The set of all rational numbers less than 0

Answer

The set of all positive rational numbers

Answer

The set of all negative rational numbers

Answer

The set of all rational numbers greater than 1

Question 4

Identify a direct implication of Lagrange's Theorem.

Accepted Answer

The order of every subgroup of a group evenly divides the order of the group

Answer

The order of a group equals the sum of the orders of its subgroups

Answer

The order of a cyclic group is always a prime number

Answer

The number of subgroups in a group is always finite

Question 5

What defines a subgroup within a group?

Accepted Answer

A non-empty subset closed under the group operation and containing the identity element.

Answer

A subset that does not form a group under the group operation.

Answer

A subset sharing some, but not all, group properties.

Answer

A subset with a different identity element than the original group.

Question 6

Let H be a subgroup of a group G. Which of the following is NOT a left coset of H in G?

Accepted Answer

G

Answer

aH

Answer

Ha

Answer

H

Question 7

Which of the following is a necessary condition for a non-empty subset H of a group G to be a subgroup of G?

Accepted Answer

Closure under the group operation

Answer

Contains the identity element

Answer

Associativity within H

Answer

Inverse elements are also in H

Question 8

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

Accepted Answer

The left and right cosets of H in G are equal

Answer

H is always abelian

Answer

G/H is always a simple group

Answer

H is the unique normal subgroup of G

Question 9

Which of the following is an example of a finite simple group?

Accepted Answer

The alternating group A5

Answer

The symmetric group S3

Answer

The cyclic group Z6

Answer

The dihedral group D4

Question 10

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

Accepted Answer

H ∩ K is a subgroup of G

Answer

H ∪ K is a subgroup of G

Answer

HK is a subgroup of G

Answer

H × K is a subgroup of G

Question 11

Which of the following is a necessary and sufficient condition for a subgroup H of G to be a maximal subgroup?

Accepted Answer

H is a proper subgroup of G and there is no subgroup of G that properly contains H

Answer

H has the same order as G

Answer

H is normal in G

Answer

H is generated by a single element

Question 12

Let G be a finite group and H be a subgroup of G. If the order of H is a prime number, then:

Accepted Answer

H is always a normal subgroup of G

Answer

H is never a normal subgroup of G

Answer

H is a normal subgroup of G if and only if G is abelian

Answer

H is a normal subgroup of G if and only if the order of G is also a prime number

Question 13

Which of the following is a necessary condition for a subgroup to be a normal subgroup?

Accepted Answer

Every left coset is also a right coset

Answer

It is a subgroup of the center of the group

Answer

It is a maximal subgroup

Answer

Its order is equal to the order of the group

Question 14

Given a group G and a subset H, which of the following conditions is sufficient to ensure that H is a subgroup of G?

Accepted Answer

H is non-empty and closed under the group operation

Answer

H contains the identity element

Answer

H is closed under inverses

Answer

H is a normal subgroup of G

Question 15

Which theorem states that the order of any subgroup divides the order of the group?

Accepted Answer

Lagrange's Theorem

Answer

Cayley's Theorem

Answer

Sylow's Theorem

Answer

Fermat's Little Theorem

Question 16

Consider a group G with the property that every element of G has order 2. What is the type of G?

Accepted Answer

Abelian

Answer

Cyclic

Answer

Simple

Answer

Solvable

Question 17

Consider a group G and a subgroup H. If H is a normal subgroup, what is the relationship between the quotient group G/H and the subgroup H?

Accepted Answer

H is isomorphic to G/H

Answer

G/H is isomorphic to H

Answer

G/H is a subgroup of H

Answer

H is a subgroup of G/H

Question 18

Consider the statement: 'Every subgroup of a finite group must be finite'. Is this statement true or false?

Accepted Answer

True

Answer

False

Question 19

In abstract algebra, factor groups play a crucial role. Explain how factor groups are constructed, describe their key properties, and discuss their significance in simplifying group structures.

Accepted Answer

Factor groups are constructed by dividing a group by a normal subgroup, preserving essential group properties.

Answer

Factor groups are irrelevant in abstract algebra, as they do not provide any meaningful insights into group structures.

Answer

Factor groups are always isomorphic to the original group, providing no additional information about the group structure.

Answer

Factor groups are only useful for studying finite groups and have no applications in infinite group theory.

Question 20

Given a group G and an element a within G, if the order of a is n, what is the order of the cyclic subgroup generated by a?

Accepted Answer

n

Answer

a

Answer

n^2

Question 21

Which of the following is a crucial condition for a subset of a group to qualify as a subgroup?

Accepted Answer

Closure under group operation

Answer

Contains the identity element

Answer

Inverse element exists for each element

Answer

Finite number of elements

Question 22

Which of these groups has precisely 3 subgroups?

Accepted Answer

Cyclic group of order 4

Answer

Cyclic group of order 6

Answer

Dihedral group of order 8

Answer

Quaternion group

Question 23

Identify an example of a normal subgroup:

Accepted Answer

Set of all invertible elements in a group

Answer

Set of even integers under addition

Answer

Set of all permutations of 3 elements

Answer

Set of all positive real numbers under multiplication

Question 24

Given a group G and a normal subgroup N, what is the definition of the coset gN of G with respect to N?

Accepted Answer

Set of elements in G equivalent to N under conjugation

Answer

Set of elements in G commuting with N

Answer

Set of elements in G that are conjugates of N

Answer

Set of subgroups of G containing N

Question 25

Which theorem asserts that the order of a subgroup divides the order of the group it belongs to?

Accepted Answer

Lagrange's Theorem

Answer

Cayley's Theorem

Answer

Sylow's Theorem

Answer

Fundamental Theorem of Algebra

Question 26

Which subset of the symmetric group S3 is not a subgroup?

Accepted Answer

{(), (1 2 3), (1 3 2)}

Answer

{(), (1 2), (1 3), (2 3)}

Answer

{(), (1 2), (1 3)}

Answer

{(), (1 2), (2 3)}

Question 27

Given subgroups H and K of a group G, what is the order of their intersection H ∩ K?

Accepted Answer

|H ∩ K|

Answer

|H| + |K| - |H ∩ K|

Answer

|H| - |H ∩ K|

Answer

|K| - |H ∩ K|

Question 28

Which of the following is a necessary condition for a non-empty subset of a group to be a subgroup?

Accepted Answer

It is closed under the group operation.

Answer

It contains the identity element.

Answer

It is a subset of the group's generator set.

Answer

It has the same order as the group.

Question 29

If a group G has finite order N, and H is a subgroup of G, then what is the maximum possible order of H?

Accepted Answer

N

Answer

N/2

Answer

N+1

Answer

N-1

Question 30

The intersection of any two subgroups of a group G is:

Accepted Answer

Always a subgroup of G

Answer

Sometimes a subgroup of G

Answer

Never a subgroup of G

Answer

Only a subgroup if the two original subgroups are normal subgroups

Question 31

Let G be a group and H a subgroup of G. The number of left cosets of H in G is equal to:

Accepted Answer

|G|/|H|

Answer

|H|/|G|

Answer

|G x H|

Answer

|H x G|

Question 32

How does Lagrange's Theorem contribute to the comprehension of finite group structure?

Accepted Answer

Lagrange's Theorem establishes a valuable relationship between the orders of finite groups and their subgroups, providing insights into group structure.

Answer

Lagrange's Theorem allows the construction of all subgroups for a given finite group, aiding in the examination of group properties.

Answer

Lagrange's Theorem directly aids in classifying groups as cyclic or non-cyclic, providing a comprehensive understanding of group structure.

Answer

Lagrange's Theorem is not essential for comprehending group structures and holds no particular significance in abstract algebra.

Question 33

Which of the following statements about subgroups is NOT true?

Accepted Answer

Subgroups always have the same order as the original group.

Answer

Subgroups are non-empty subsets of the original group.

Answer

Subgroups are closed under the group operation.

Answer

Subgroups contain the identity element of the original group.

Question 34

According to Lagrange's Theorem, what is true about the order of a subgroup?

Accepted Answer

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

Answer

It is always equal to the order of the group.

Answer

It can be any positive integer.

Question 35

Which of the following is a necessary and sufficient condition for a non-empty subset of a group G to be a subgroup of G?

Accepted Answer

It is closed under the group operation and contains the identity element

Answer

It contains the identity element

Answer

It is a non-empty subset of G

Answer

It is closed under inverses

Question 36

According to Lagrange's Theorem, the order of a subgroup H of a finite group G must be:

Accepted Answer

A divisor of the order of G

Answer

Equal to the order of G

Answer

Greater than the order of G

Answer

Less than the order of G

Question 37

Which of the following is NOT a property of subgroups?

Accepted Answer

They are always normal subgroups

Answer

They are closed under inverses

Answer

They contain the identity element

Answer

They are closed under the group operation

Question 38

Which of the following statements about the intersection of two subgroups of a group G is true?

Accepted Answer

It is always a subgroup of G

Answer

It is a subgroup of G if and only if the subgroups are abelian

Answer

It is a subgroup of G if and only if the subgroups are normal

Question 39

If a group G has a proper subgroup H, the quotient group G/H is:

Accepted Answer

A group

Answer

A cyclic group

Answer

A field

Answer

An abelian group

Question 40

Which of the following is a direct consequence of the fact that a subgroup of a finite group is finite?

Accepted Answer

The order of the subgroup divides the order of the group

Answer

The group is abelian

Answer

The subgroup is normal

Question 41

Let G be the group of integers under addition, denoted (Z, +). Which of the following subsets of Z is a subgroup of G?

Accepted Answer

The set of even integers

Answer

The set of odd integers

Answer

The set of prime numbers

Question 42

According to Lagrange's Theorem, if G is a finite group of order n and H is a subgroup of G, what can be said about the order of H?

Accepted Answer

The order of H, denoted |H|, must divide the order of G, |G|.

Answer

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

Answer

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

Question 43

In the context of group theory, what is the definition of the left coset of a subgroup H in a group G with respect to an element g?

Accepted Answer

The left coset of H with respect to g, denoted by gH, is the set of all elements formed by multiplying g with every element in H.

Answer

The left coset of H with respect to g is the intersection of H and the cyclic subgroup generated by g.

Answer

The left coset of H with respect to g is the set of all elements that commute with g within H.

Question 44

Let G be a finite group and H be a subgroup of G. What is the significance of the number of distinct left cosets of H in G?

Accepted Answer

The number of distinct left cosets of H in G is equal to the index of H in G, which is denoted as [G:H] and represents the number of left cosets.

Answer

The number of distinct left cosets of H in G is always equal to the order of G.

Answer

The number of distinct left cosets of H in G is always greater than the order of H.

Question 45

Consider a group G and two subgroups H and K. Which of the following statements is true about the intersection of H and K?

Accepted Answer

The intersection of H and K, denoted as H ∩ K, is also a subgroup of G.

Answer

The intersection of H and K is a subgroup of H but not necessarily of G.

Answer

The intersection of H and K is a subgroup of G only if H and K are normal subgroups of G.

Question 46

Which of the following is a crucial requirement for a non-empty subset of a group to be considered a subgroup?

Accepted Answer

The subset must be closed under the group operation and contain the identity element.

Answer

The subset must contain the inverse of every element in the group.

Answer

The subset must have the same order as the group.

Question 47

Provide an example of a subgroup of the group of integers under addition, (Z, +).

Accepted Answer

The set of even integers, denoted as 2Z, is a subgroup of (Z, +).

Answer

The set of odd integers is a subgroup of (Z, +).

Answer

The set of prime numbers is a subgroup of (Z, +).

Question 48

According to Lagrange's Theorem, let G be a finite group of order n and H be a subgroup of G. What can be concluded about the possible orders of H?

Accepted Answer

The order of H, denoted as |H|, must be a divisor of the order of G, |G|.

Answer

The order of H is always greater than the order of G.

Answer

The order of H can be any integer less than the order of G.

Question 49

Which of the following statements about normal subgroups is true?

Accepted Answer

A normal subgroup is invariant under conjugation.

Answer

A normal subgroup is always a proper subgroup.

Answer

A subgroup is normal if and only if it is abelian.

Question 50

Let G be a group and let H and K be subgroups of G. Which of the following is always a subgroup of G?

Accepted Answer

H ∩ K

Answer

HK

Answer

G/H

Answer

H ∪ K

Question 51

Which of the following is a type of subgroup that is characterized by its central elements?

Accepted Answer

Center

Answer

Normalizer

Answer

Quotient group

Answer

Coset

Question 52

Let G be a group and let a be an element of G. The set of all elements of G that commute with a forms a subgroup of G called the:

Accepted Answer

Centralizer of a

Answer

Conjugacy class of a

Answer

Normalizer of a

Answer

Stabilizer of a

Question 53

Let G be a finite group and let p be a prime number. If p divides the order of G, then G must have a subgroup of order:

Accepted Answer

p

Answer

2p

Answer

p^2

Answer

It cannot be determined from the given information.