Is this an isomoprphism of groups?

G is the group of 2x2 invertible upper triangular matrices with real entries. Let H be the normal subgroup such that .

If I have the following surjective homomorphism

such that .

Then by the first isomorphism theorem I have the following isomorphism:

such that

Is this correct?

Thanks for any help

Re: Is this an isomoprphism of groups?

Re: Is this an isomoprphism of groups?

It's correct. Maybe you should denote by the morphism between (I guess is the matrix of which have determinant ) and .

Re: Is this an isomoprphism of groups?

yeah thanks very much, I edited it. My notation is bad, how did you put the squiggle above the f?

Re: Is this an isomoprphism of groups?

Quote:

Originally Posted by

**hmmmm** yeah thanks very much, I edited it. My notation is bad, how did you put the squiggle above the f?

\widetilde f

Re: Is this an isomoprphism of groups?

Uh,

I don't think so. The kernel of that homomorphism is certainly strictly larger than . What about the matrix ? Certainly but .

Re: Is this an isomoprphism of groups?

Ah of course, in fact if then that maps to the identity.

So i have to have another map f such that and it has to be surjective right?

Re: Is this an isomoprphism of groups?

What about:

Let J be the subgroup of such that .

Then we have a surjectrive homomorphism [\bar{f}:G\rightarrow J[/TEX] with kernel H.

So from the first isomorphism theorem we have an isomorphism:

thanks for any help

Re: Is this an isomoprphism of groups?

Quote:

Originally Posted by

**hmmmm** What about

:

Let J be the subgroup of

such that

.

Then we have a surjectrive homomorphism [\bar{f}:G\rightarrow J[/TEX] with kernel H.

So from the first isomorphism theorem we have an isomorphism:

thanks for any help

What precisely are you trying to do? Find a nicer description of ? Is this a different question?

Re: Is this an isomoprphism of groups?

Yeah that is what I am trying to do, any help would be apperciated