Does φ(N) Form a Normal Subgroup in G' When φ is a Homomorphism?

  • Thread starter ThatPinkSock
  • Start date
  • Tags
    Proof
In summary, we need to prove that if φ is a homomorphism of G onto G' and N is a normal subgroup of G, then φ(N) is also a normal subgroup of G. To do this, we can use the fact that φ is onto, meaning that every element in G' is the image of some element in G. Then, by using the properties of homomorphisms, we can show that kφ(n)k-1 = φ(N), which proves that φ(N) is indeed a normal subgroup of G.
  • #1
ThatPinkSock
3
0
From Herstein's Abstract Algebra. Section 2.7 #7



If φ is a homomorphism of G onto G' and N ◅ G, show that φ(N) ◅ G.



Attempt:
I want to prove that if k ∈ G' then kφ(N)k-1 = φ(N), but k = φ(n) for some n... then idk what.
 
Physics news on Phys.org
  • #2
Take a in N, you need to prove that

[tex]k\varphi(a)k^{-1}\in \varphi(N)[/tex]

Replace k with [itex]\varphi(n)[/itex], what do you get??
 
  • #3
the fact that φ is onto is important.

this means that EVERY k in G' is the image of some g in G:

k = φ(g). now use the fact that φ is a homomorphism.

what can we say about kφ(n)k-1?
 

FAQ: Does φ(N) Form a Normal Subgroup in G' When φ is a Homomorphism?

What is a homomorphism?

A homomorphism is a function between two algebraic structures that preserves the operations of the structures. In other words, it maps elements from one structure to another in a way that respects the algebraic operations.

What is the significance of Homomorphism Proof in mathematics?

Homomorphism Proof is significant in mathematics because it allows for the study of the relationships between different algebraic structures. It also helps to establish a connection between seemingly unrelated structures and can reveal hidden similarities and patterns.

How is Homomorphism Proof used in abstract algebra?

In abstract algebra, Homomorphism Proof is used to show that a function between two algebraic structures is a homomorphism. This is useful in proving theorems and properties of algebraic structures, such as groups, rings, and fields.

What is the process of proving a function is a homomorphism?

The process of proving a function is a homomorphism involves showing that the function preserves the algebraic operations of the structures. This is typically done by showing that the function satisfies the necessary properties, such as the preservation of identity and the operation of the structures.

What are some applications of Homomorphism Proof in other fields?

Homomorphism Proof has applications in various fields, such as computer science, cryptography, and physics. In computer science, it is used in the design and analysis of algorithms. In cryptography, it is used to ensure the security of encrypted messages. In physics, it is used in the study of symmetries and conservation laws.

Similar threads

MHB 4 part problem regarding homomorphisms
Replies
1
Views
1K
MHB Can You Prove These Properties of Normal Subgroups in Group Theory?
Replies
2
Views
2K
Proving Normality in G: An Elegant Approach?
Replies
6
Views
2K
B Is it invalid to redefine the sgn function in this way in a proof?
Replies
15
Views
4K
MHB Show that G is nilpotent when the quotient group is nilpotent
Replies
1
Views
1K
MHB G contains a normal p-Sylow subgroup
Replies
5
Views
2K
Proving that Aut(G) is a subgroup of Bij(G)
Replies
6
Views
2K
A How many cosets are there when taking a subgroup in a group and forming cosets?
Replies
1
Views
1K
Showing that a certain subgroup is the kernal of a homomorphism
Replies
9
Views
4K
I What is the meaning of N(H) in subgroup notation?
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top