# First Isomorphism Theorem

## Homework Statement

Suppose H is a normal subgroup G and L is a subgroup of K. Then (G x K)/(H x L) is isomorphic to (G/H) x (K/L)

## The Attempt at a Solution

I know that I have to use the First Isomorphism Theorem, but in order to do that I need some function phi. I am having a really difficult time finding a function from (G x K) to (G/H)x(K/L). If I have this I am almost certain I can complete the proof with the First Isomorphism Theorem.

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Stephen Tashi
Does the problem say "L is a subgroup of K"? It's not necessarily a normal subgroup, correct?

It says that L is a normal subgroup of K.

Stephen Tashi
It says that L is a normal subgroup of K.
That's good. You have already studied the picture where there is homomorphism from G to some other group (which could also be G) and map phi1 from G to G/H.

Livewise you can have a picture where there is a map from K to some other group and a map phi2 from K to K/L. The group G X K is just ordered pairs of elements from the two groups that look like {g, k}. You can use phi1 to map the g-element into G/H and the phi2 map to map the k-element into K/L. That defines a map from GxH to G/H x K/L.

you are allowed to use two different phis??

Stephen Tashi