2) Subgroups and Homomorphisms Flashcards
What is the trivial subgroup
In any group G with identity element e, the set {e} is a trivial subgroup of G
What is a proper subgroup
A subgroup H of G is called proper if H ≠ G
How does the identity element of a subgroup relate to the identity element of the entire group
In any subgroup H of a group G, the identity element of H is the same as the identity element of G
What is the subgroup criteria
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
Are the natural numbers a subgroup of the integers
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
What can be said about the intersection of two subgroups within the same group
The intersection of any two subgroups of the same group is itself a subgroup
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
Give an example of a subgroup of a non-abelian group that is abelian
What is the special linear group
SL(n, K) = {A ∈ GL(n, K) | det A = 1}
is a subgroup of GL(n, K)
What are the different types of subgroups of matrices commonly studied in linear algebra and group theory
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
What are the powers of a group element
What does < g > denote
{g^k | k ∈ Z}
What are the properties of powers of group elements
What type of subgroup is formed by all powers of a fixed element
Cyclic Subgroup
Describe the proof that a cyclic subgroup is an abelian subgroup
What is a cyclic subgroup
Let G be a group and a ∈ G. The subgroup < a > of
G is called the cyclic subgroup generated by a
When is a group cyclic
A group G is called cyclic, if there exists an element
a ∈ G such that G = < a >
Prove that (Q, +) is not a cyclic group
What is a generator
- 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
What is a group homomorphism
What is the relationship between isomorphisms and homomorphisms
All isomorphisms are homomorphisms, but not all homomorphisms are isomorphisms
What is the image and kernel of a homomorphism
What are the properties of homomorphisms
Describe the proof of the properties of homomorphisms
What can be said about the inverse of an isomorphism
Describe the proof that the inverse of an isomorphism is also an isomorphism
