If and K are normal subgroups of a group with , prove that .
Hint: If is defined by , then ; moreover, we have , so that .
I have managed to prove that . I'm having difficulty in proving that is surjective. This is crucial in proving the above isomorphism using 1st Isomorphism Theorem.
Hey,
The proof is as below:
We want to show that , such that .
Since , and .
Since , and .
This implies that .
Since , .
This implies that and .
Hence such that .
Therefore .
Clearly, .
Finally, we conclude that .
From there, we can establish the isomorphism using First Isomorphism Theorem.
So, is the above proof correct?
Thanks in advance.