2) Subgroups and Homomorphisms

Question 1

What is the trivial subgroup

Answer 1

In any group G with identity element e, the set {e} is a trivial subgroup of G

Question 2

What is a proper subgroup

Answer 2

A subgroup H of G is called proper if H ≠ G

Question 3

How does the identity element of a subgroup relate to the identity element of the entire group

Answer 3

In any subgroup H of a group G, the identity element of H is the same as the identity element of G

Question 4

What is the subgroup criteria

Answer 4

A non-empty subset H of G is a subgroup if and only if the following two conditions both hold -
(i) ∀g, h ∈ H we have gh ∈ H,
(ii) ∀h ∈ H we have h^-1 ∈ H

Question 5

Are the natural numbers a subgroup of the integers

Answer 5

N is not a subgroup of Z. Although the sum of two natural numbers is a natural number, the inverse n of n ∈ N is not in N

Question 6

What can be said about the intersection of two subgroups within the same group

Answer 6

The intersection of any two subgroups of the same group is itself a subgroup

Question 7

Give an example of a group G and subgroups H ≤ G, K ≤ G
such that the union H ∪ K is not a subgroup of G

Answer 7

Question 8

Give an example of a subgroup of a non-abelian group that is abelian

Answer 8

Question 9

What is the special linear group

Answer 9

SL(n, K) = {A ∈ GL(n, K) | det A = 1}
is a subgroup of GL(n, K)

Question 10

What are the different types of subgroups of matrices commonly studied in linear algebra and group theory

Answer 10

UT(n, K) - Upper Uni-Triangular Matrices
T(n,K) - Upper Triangular Matrices
SL(n, K) - Special Linear Group
D(n, K) - Invertible Diagonal Matrices
Scal(n,K) - Scalar Matrices

Question 11

What are the powers of a group element

Answer 11

Question 12

What does < g > denote

Answer 12

{g^k | k ∈ Z}

Question 13

What are the properties of powers of group elements

Answer 13

Question 14

What type of subgroup is formed by all powers of a fixed element

Answer 14

Cyclic Subgroup

Question 15

Describe the proof that a cyclic subgroup is an abelian subgroup

Answer 15

Question 16

What is a cyclic subgroup

Answer 16

Let G be a group and a ∈ G. The subgroup < a > of
G is called the cyclic subgroup generated by a

Question 17

When is a group cyclic

Answer 17

A group G is called cyclic, if there exists an element
a ∈ G such that G = < a >

Question 18

Prove that (Q, +) is not a cyclic group

Answer 18

Question 19

What is a generator

Answer 19

• An element from which every other element of the group can be derived through repeated applications of the group operation
• In cyclic groups, a single generator can produce the entire group

Question 20

What is a group homomorphism

Answer 20

Question 21

What is the relationship between isomorphisms and homomorphisms

Answer 21

All isomorphisms are homomorphisms, but not all homomorphisms are isomorphisms

Question 22

What is the image and kernel of a homomorphism

Answer 22

Question 23

What are the properties of homomorphisms

Answer 23

Question 24

Describe the proof of the properties of homomorphisms

Answer 24

Question 25

What can be said about the inverse of an isomorphism

Answer 25

Question 26

Describe the proof that the inverse of an isomorphism is also an isomorphism

Answer 26