First Isomorphism Theorem/quotient group question

**hmmmm** · July 28th 2011, 03:55 AM

Let G be the group of invertible upper triangular 2x2 matrices over the real numbers. Determine if the following are normal subgroups and if they are use the frist isomorphism theorem to identify the quetient group G/H.

where H is

a) $a_{11}=1$

b) $a_{11}=a_{22}$

I have checked (hopefully correctly) that these are both normal subgroup (via $gHg^{-1}=H$ ) However I am unsure as to what the next part is actually asking me?

Thanks for any help

**Swlabr** · July 28th 2011, 04:42 AM

Originally Posted by hmmmm

Let G be the group of invertible upper triangular 2x2 matrices. Determine if the following are normal subgroups and if they are use the frist isomorphism theorem to identify the quetient group H.

where H is

a) $a_{11}=1$

b) $a_{11}=a_{22}$

I have checked (hopefully correctly) that these are both normal subgroup (via $gHg^{-1}=H$ ) However I am unsure as to what the next part is actually asking me?

Thanks for any help

For a) the question is now wanting you to find a map from G to another group, H, such that the kernel is the set of all matrices such that $a_{11}=1$ . It is analogous for b).

If you have any trouble finding these maps, just ask. People will happily surrender the answers, but you finding them yourself is much, much more useful!

Finally, what are your matrices over? $\mathbb{Z}$ ? $\mathbb{Q}$ ? $\mathbb{R}$ ? Some arbitrary field? (The question doesn't really make sense unless you know this...)

**hmmmm** · July 28th 2011, 05:34 AM

Sorry I should have said that it is over the real numbers I have edited it now.

Im a bit confused by your answer I want to find an isomorphism from G to H (H being a subgroup of G this isnt possible is it?) or do you just mean a different group H'

Thanks for the help sorry for my confusion

for a

So am I looking for a map $\phi:G\rightarrow G'$ such that $\phi(g)\rightarrow a_{11}$ and where $G'=(\mathbb{R^*},\times)$ ?

So the quotient group G/H is isomorphic to $\mathbb{R^*},\times)$

**Swlabr** · July 28th 2011, 11:58 AM

Originally Posted by hmmmm

Sorry I should have said that it is over the real numbers I have edited it now.

Im a bit confused by your answer I want to find an isomorphism from G to H (H being a subgroup of G this isnt possible is it?) or do you just mean a different group H'

Thanks for the help sorry for my confusion

for a

So am I looking for a map $\phi:G\rightarrow G'$ such that $\phi(g)\rightarrow a_{11}$ and where $G'=(\mathbb{R^*},\times)$ ?

So the quotient group G/H is isomorphic to $\mathbb{R^*},\times)$

Yeah, sorry, I just meant a different group. The isomorphism looks correct, and would be my first guess, but you should check first that it is well-defined, surjective, and that the kernel is what you want it to be...

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