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

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• October 10th 2011, 03:44 AM
Bernhard
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 $GL_n$( $\mathbb{C}$) isomorphic to a subgroup of $GL_ {2n}$( $\mathbb{R}$)?

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

Peter
• October 10th 2011, 04:08 AM
Swlabr
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 $GL_n$( $\mathbb{C}$) 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"...
• October 10th 2011, 04:56 AM
Bernhard
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

$\phi$: (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
• October 10th 2011, 05:06 AM
Swlabr
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

$\phi$: (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 $\phi: (A+iB) \mapsto \left( \begin{array}{cc} A & B\\-B & A\end{array} \right)$...
• October 10th 2011, 05:19 AM
Bernhard
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
• October 10th 2011, 01:29 PM
Drexel28
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 $GL_n$( $\mathbb{C}$) 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 $\text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$.
• October 10th 2011, 02:34 PM
Deveno
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Quote:

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?
• October 10th 2011, 03:06 PM
Drexel28
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Quote:

Originally Posted by Deveno
do you mean GL:Vect-->Grp is a functor?

You bet I do.