General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups

**Bernhard** · October 10th 2011, 03:44 AM

Can anyone help with the following problem from Artin - Algebra Ch9 on Linear Groups.

Is $GL_n$ ( $\mathbb{C}$ ) isomorphic to a subgroup of $GL_ {2n}$ ( $\mathbb{R}$ )?

How do I approach proving this one way or the other?

Peter

**Swlabr** · October 10th 2011, 04:08 AM

Originally Posted by Bernhard

Can anyone help with the following problem from Artin - Algebra Ch9 on Linear Groups.

Is $GL_n$ ( $\mathbb{C}$ ) isomorphic to a subgroup of $GL_ {2n}$ ( $\mathbb{R}$ )?

How do I approach proving this one way or the other?

Peter

Start with $n=1$ , and see where that takes you. Can you find a 2x2 real matrix which squares to -I, where I is the 2x2 real identity matrix.

One you have found this matrix, you want to "expand" $GL_n(\mathbb{R})$ by this matrix. Try and work out what I mean by "expand"...

**Bernhard** · October 10th 2011, 04:56 AM

Taking your advice, it looks like something like the following may work for n = 1

$\phi$ : (a + ib) $\rightarrow$ $\left(\begin{array}{cc}a&b\\-b&a\end{array}\right)$ :

should be OK for n = 1

But how to 'expand' $GL_{2n}$ ( $\mathbb{R}$ )???

Can you help?

Peter

**Swlabr** · October 10th 2011, 05:06 AM

Originally Posted by Bernhard

Taking your advice, it looks like something like the following may work for n = 1

$\phi$ : (a + ib) $\rightarrow$ $\left(\begin{array}{cc}a&b\\-b&a\end{array}\right)$ :

should be OK for n = 1

But how to 'expand' $GL_{2n}$ ( $\mathbb{R}$ )???

Can you help?

Peter

Let A and B be nxn matrices. Then let $\phi: (A+iB) \mapsto \left( \begin{array}{cc} A & B\\-B & A\end{array} \right)$ ...

**Bernhard** · October 10th 2011, 05:19 AM

Will check this out!

Thanks so much for the help. Appreciate your assistance!

Will now try to go further with Artin's chapter on the Linear Groups!

Peter

**Drexel28** · October 10th 2011, 01:29 PM

Originally Posted by Bernhard

Can anyone help with the following problem from Artin - Algebra Ch9 on Linear Groups.

Is $GL_n$ ( $\mathbb{C}$ ) isomorphic to a subgroup of $GL_ {2n}$ ( $\mathbb{R}$ )?

How do I approach proving this one way or the other?

Peter

Perhaps a more conceptual way of looking at it, if that kind of thing makes you happy, is that if $V,W$ are isomorphic vector spaces then $\text{GL}(V),\text{GL}(W)$ are isomorphic groups. Now, evidently $\dim_\mathbb{R}\mathbb{C}^n=2n$ so that $\mathbb{C}^n\cong\mathbb{R}^{2n}$ as real vector spaces, and so $\text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$ .

**Deveno** · October 10th 2011, 02:34 PM

Originally Posted by Drexel28

Perhaps a more conceptual way of looking at it, if that kind of thing makes you happy, is that if $V,W$ are isomorphic vector spaces then $\text{GL}(V),\text{GL}(W)$ are isomorphic groups. Now, evidently $\dim_\mathbb{R}\mathbb{C}^n=2n$ so that $\mathbb{C}^n\cong\mathbb{R}^{2n}$ as real vector spaces, and so $\text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$ .

do you mean GL:Vect-->Grp is a functor?

**Drexel28** · October 10th 2011, 03:06 PM

Originally Posted by Deveno

do you mean GL:Vect-->Grp is a functor?

You bet I do.

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