How Can You Verify the Definitions of Homomorphism and Subgroup?

  • Thread starter Monkeyfry180
  • Start date
In summary, the conversation discusses the definitions of "homomorphism" and "subgroup" and how they relate to group operations. It is determined that for a subgroup to be valid, it must contain the 0 element and be closed under the group operation. The question is then posed whether the 0 function is in the subgroup and if the addition of two functions in the subgroup will also result in a function in the subgroup.
  • #1
Monkeyfry180
7
0
Gracias
 
Last edited:
Physics news on Phys.org
  • #2


Okay, what have you tried? This really only involves verifying the definitions of "homomorphism" and "subgroup".

The only condition on a "homomorphism" is that it maintain group operations. If f and g are two differentiable functions, is it true that theta(f+ g)= theta(f)+ theta(g)?

Since saying that C1 is a group means that things like associativity of the operation is true, we don't have to reprove those for operations in the subgroup. We only need to show
(1) the 0 element is in the subset
(2) the subgroup is "closed" under the group operation

The 0 element of this group is the 0 function, f(x)= 0 for all x. Is that in this subset?

Suppose f and g are in this subset. Is f+ g also in this subset? That is, is (f+ g)(0)= 0?
 

1. What is a homomorphism?

A homomorphism is a mathematical concept that describes a relationship between two mathematical structures, where one structure can be mapped onto the other structure in a way that preserves the operations and relationships of the original structure.

2. How is a homomorphism different from an isomorphism?

A homomorphism is a weaker version of an isomorphism. While an isomorphism preserves both the operations and relationships of the original structure, a homomorphism only preserves the operations.

3. What is the significance of Gracias Homomorphism?

Gracias Homomorphism is a specific type of homomorphism that has been extensively studied in mathematics and has applications in various fields such as computer science, physics, and economics. It has been used to study symmetry, patterns, and relationships between different mathematical structures.

4. How can Gracias Homomorphism be applied in real-world problems?

Gracias Homomorphism has been applied in various real-world problems, such as data compression, graph theory, and cryptography. It has also been used in the development of algorithms and computer programs.

5. Is there any practical use for understanding Gracias Homomorphism?

Yes, understanding Gracias Homomorphism can have practical applications in various fields, including data analysis, artificial intelligence, and secure communication systems. It can also help in understanding and solving complex mathematical problems in a more efficient and systematic way.

Similar threads

MHB 4 part problem regarding homomorphisms
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Exploring the Significance of p-Sylow Subgroups in Group Theory
    • Linear and Abstract Algebra
Replies
2
Views
820
I Proof for subgroup -- How prove it is a subgroup of Z^m?
    • Linear and Abstract Algebra
Replies
11
Views
1K
I Terminology involving "homomorphic"
    • Linear and Abstract Algebra
Replies
19
Views
2K
MHB Number of Homomorphisms from $\mathbb{Z}_4$ to $S_4$: A Brief Exploration
    • Linear and Abstract Algebra
Replies
11
Views
2K
MHB Can You Prove These Properties of Normal Subgroups in Group Theory?
    • Linear and Abstract Algebra
Replies
2
Views
1K
A Does this construction verify the universal property?
    • Linear and Abstract Algebra
Replies
7
Views
909
MHB 4 problems regarding automorphisms/homomorphisms
    • Linear and Abstract Algebra
Replies
5
Views
2K
I Understand homomorphisms from Z^a --> Z^b
    • Linear and Abstract Algebra
Replies
1
Views
1K
Preimage of arbitrary subgroup.
    • Linear and Abstract Algebra
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top