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

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

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

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

Peter

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

Quote:

Originally Posted by

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

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

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

Peter

Start with $\displaystyle n=1$, 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" $\displaystyle GL_n(\mathbb{R})$ by this matrix. Try and work out what I mean by "expand"...

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

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

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

should be OK for n = 1

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

Can you help?

Peter

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

Quote:

Originally Posted by

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

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

should be OK for n = 1

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

Can you help?

Peter

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

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

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

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

Quote:

Originally Posted by

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

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

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 $\displaystyle V,W$ are isomorphic vector spaces then $\displaystyle \text{GL}(V),\text{GL}(W)$ are isomorphic groups. Now, evidently $\displaystyle \dim_\mathbb{R}\mathbb{C}^n=2n$ so that $\displaystyle \mathbb{C}^n\cong\mathbb{R}^{2n}$ as real vector spaces, and so $\displaystyle \text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$.

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

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**Drexel28** Perhaps a more conceptual way of looking at it, if that kind of thing makes you happy, is that if $\displaystyle V,W$ are isomorphic vector spaces then $\displaystyle \text{GL}(V),\text{GL}(W)$ are isomorphic groups. Now, evidently $\displaystyle \dim_\mathbb{R}\mathbb{C}^n=2n$ so that $\displaystyle \mathbb{C}^n\cong\mathbb{R}^{2n}$ as real vector spaces, and so $\displaystyle \text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$.

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

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

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

You bet I do.