Thread: Group Theory - newcomer needs help

Given that A = {}

Define * on A by (p,q) * (r,s) = (pr+qs, ps+qr)

Does anyone know what an identity element for * on A might be? And how do I show that (A,*) is a group other than having shown that it is associative and is closed under *?

Thank you in advance,
Aalborg

Originally Posted by Alborg

Given that A = {}

Define * on A by (p,q) * (r,s) = (pr+qs, ps+qr)

Does anyone know what an identity element for * on A might be? And how do I show that (A,*) is a group other than having shown that it is associative and is closed under *?

Thank you in advance,
Aalborg

I think is an identity element for * on A.

Hello,

For the identity element, I think that you have to solve for p and q in :

Hi

And how do I show that (A,*) is a group other than having shown that it is associative and is closed under *?

You also need show that there exists such that for all pairs . (you can, for example, solve for and then deduce the equality )

( : identity element)

Originally Posted by flyingsquirrel

Hi

You also need show that there exists such that for all pair . (you can, for example, first solve for and the equality will come after)

( : identity element)

Given let be defined as such:

If , then take such that .

If , then take such that .

This shows that every element has an inverse under the identity .