# Homomorphism

#### Lauren1231

I know the formula that I have to show is true. However, I am confused where to start to prove it. I thought multiplying the 2 matrices together but that’s not possible. Can someone start me off please?

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What is P?

#### Greens

I’m out of the house at the moment but I’ll give some tips to get you started.

1. Show that the elements of $P$ form a group under addition. (if it’s not given that it’s a group). $GL(2 , \mathbb{F})$ forms a group under matrix multiplication.

2. To show $\phi$ is a group homomorphism, show that for any two elements $p, q \in P$

$\phi (p+q) = \phi(p) \cdot \phi(q)$

3. Show that each element of $GL$ is mapped to, keep in mind that $P$ defines that $dg-ef \neq 0$ remember the meaning of that.

4. Keeping in mind that all elements of $GL$ are invertible, what would the null element be?

Lauren1231

#### Lauren1231

I’m out of the house at the moment but I’ll give some tips to get you started.

1. Show that the elements of $P$ form a group under addition. (if it’s not given that it’s a group). $GL(2 , \mathbb{F})$ forms a group under matrix multiplication.

2. To show $\phi$ is a group homomorphism, show that for any two elements $p, q \in P$

$\phi (p+q) = \phi(p) \cdot \phi(q)$

3. Show that each element of $GL$ is mapped to, keep in mind that $P$ defines that $dg-ef \neq 0$ remember the meaning of that.

4. Keeping in mind that all elements of $GL$ are invertible, what would the null element be?
Got you. I may need some additional help when you have time, but I’ll give it a go and show my workings. Also, what does phi actually stand for? For section 2?

#### Greens

It’s the function, just a different styling of letter.

#### Lauren1231

It’s the function, just a different styling of letter.
Oh so am I wrong in finding the determinate of p+q? Or is that my function

#### Greens

That's the function. The definition of the function is given in the image.

#### Lauren1231

That's the function. The definition of the function is given in the image.
Got you and can I chose an example like this or would I need to do it using letters?
(Sorry I’m struggling to get my head around this)

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#### Greens

Have to use letters. $p$ and $q$ have to represent ANY element of $P$. If you use examples you can show it, sure, but all that means is that the proof holds for those examples. It means nothing for the entire group.

Also, use the function symbol given to you so you avoid confusing the grader, if there is one. $\varphi$

Lauren1231