# Using the Second Isomorphism (Diamond Isomorphism) Theorem

DeldotB

## Homework Statement

Good day all,

Im completely stumped on how to show this:

|AN|=(|A||N|/A intersect N|)

Here: A and N are subgroups in G and N is a normal subgroup.
I denote the order on N by |N|

## Homework Equations

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Second Isomorphism Theorem

## The Attempt at a Solution

Well, I know know I am supposed to use the theorem, but I have no idea where to start!
I don't seem how to relate what the theorem says to orders of sets...

Any help would be greatly appreciated!

Homework Helper
Gold Member
First, to be able to use the 2nd isomorphism theorem at all, you need to prove that AN is a subgroup. Has that been given to you as a theorem? If not, can you prove it (it's quite easy)?

Next note that the problem statement only makes sense if A and N are finite.

With those out of the way, the second isomorphism theorem says:

$$AN / N\cong A/A\cap N$$
from which it follows that
$$|AN / N|=| A/A\cap N|$$
You have been asked to prove

$$\frac{|AN|}{|N|}=\frac{|A|}{|A\cap N|}$$

So if you can prove that, for any finite group ##G## and normal subgroup ##N##,
$$|G/N| =\frac{|G|}{|N|}$$
then you're almost done.

Can you prove that? Think about the size ##|gN|## of each coset ##gN##.

DeldotB
Ahh! Thanks! I didnt realize I could just divide by |N|...I am new to quotient groups and for some reason I havnen't been able to get the hang of them yet. Thanks!

I'll use Lagranges Theorem for the last part