2) Subgroups and Homomorphisms Flashcards
(26 cards)
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
