Is f a Surjective and Injective Isomorphism from HxN to HN in G?

  • Thread starter Thread starter mathmajor2013
  • Start date Start date
  • Tags Tags
    Isomorphism
Click For Summary

Homework Help Overview

The discussion revolves around proving that a function f defined from the direct product of two normal subgroups H and N of a group G to the subgroup HN of G is an isomorphism. Participants are exploring the properties of this function, particularly focusing on its homomorphic nature and the implications of the intersection of the subgroups.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to establish whether f is a homomorphism by verifying the property f((h,n)(h',n'))=f((hh',nn')). There is also discussion about using the pigeonhole principle to argue about the sizes of HxN and HN, given that their intersection is trivial.

Discussion Status

The conversation is ongoing, with some participants confirming steps related to the homomorphism property while others are questioning the best approach to demonstrate that f is surjective and injective. There is a recognition of the complexity in showing that the two groups are of the same size.

Contextual Notes

Participants are navigating the constraints of the problem, including the requirement to show both injectivity and surjectivity without assuming prior knowledge of isomorphism properties. The nature of the subgroups and their intersection is also a focal point of discussion.

mathmajor2013
Messages
26
Reaction score
0
Let G be a group, H a normal subgroup, N a normal subgroup, and H intersect N = {e}. Let H x N be the direct product of H and N. Prove that f: HxN->G given by f((h,n))=hn is an isomorphism from HxN to the subgroup HN of G.
Hint: For all h in H and n in N, hn=nh.
 
Physics news on Phys.org
This looks a lot like a homework/coursework question. This really should belong there instead of the main forums...

To the point: what have you tried already?
 
I am confused how to start this problem. To first show it is a homomorphism, is f((h,n)(h',n'))=f((hh',nn'))?
 
mathmajor2013 said:
I am confused how to start this problem. To first show it is a homomorphism, is f((h,n)(h',n'))=f((hh',nn'))?

Yes, that step is already correct. Now apply the definition of f...
 
Right the homomorphism part is easy now. Am I able to use the pigeonhole principle for the isomorphic part? That is, are HxN and HN the same size? It seems like they are since H intersect N is only the identity.
 
It might be tricky to see that they are thesame size. Isn't easier to show that f is surjective and injective?
 

Similar threads

Having all subgroups normal is isomorphism invariant
  • · Replies 1 ·
Replies
1
Views
2K
Is N Always in the Center of G or Does It Intersect H or K Nontrivially?
  • · Replies 7 ·
Replies
7
Views
3K
Subgroup of Index ##n## for every ##n \in \Bbb{N}##.
  • · Replies 1 ·
Replies
1
Views
2K
Showing that Q_8 can't be written as a direct product
  • · Replies 11 ·
Replies
11
Views
4K
Isomorphism between HK and H x K
  • · Replies 19 ·
Replies
19
Views
5K
Normal group of order 60 isomorphic to A_5
  • · Replies 6 ·
Replies
6
Views
2K
Abstract Algebra; Group Theory Question
  • · Replies 1 ·
Replies
1
Views
1K
Do Isomorphic Groups Have Isomorphic Centers?
  • · Replies 3 ·
Replies
3
Views
2K
Showing that every finite group has a composition series
  • · Replies 3 ·
Replies
3
Views
3K
An exercise with the third isomorphism theorem in group theory
  • · Replies 5 ·
Replies
5
Views
3K