Show they are isomorphic

  • Thread starter symbol0
  • Start date
77
0
Let M and N be normal subgroups of G such that G=MN.
Prove that G/(M[itex]\bigcap[/itex]N)[itex]\cong[/itex](G/M)x(G/N).

I tried coming up with an isomorphism from G to (G/M)x(G/N) such that the kernel is M[itex]\bigcap[/itex]N, so that I can use the fundamental homomorphism theorem.
I tried f(a) = (aM, aN). It is an homomorphism and M[itex]\bigcap[/itex]N is the kernel but I'm having a hard time showing it is onto.

I would appreciate any help.
Thank you
 

CompuChip

Science Advisor
Homework Helper
4,284
47
So you need to show that for all g, g' in G, there is an a in G such that: (g M, g' N) = (a M, a N).

I haven't worked this out in detail, but: since G = MN, you can write g = m n, g' = m' n'. I suspect that a = m' n might do the trick.
You will need that M and N are normal, so in particular h M = M h, h N = N h for all h in G.
 
77
0
Thank you Compuchip
 

Related Threads for: Show they are isomorphic

Replies
32
Views
20K
  • Posted
Replies
4
Views
3K
  • Posted
Replies
1
Views
2K
Replies
3
Views
522
Replies
1
Views
548
Replies
11
Views
15K
Replies
1
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top