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

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    Super Member Bernhard's Avatar
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    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
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    MHF Contributor Swlabr's Avatar
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    Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups

    Quote Originally Posted by Bernhard View Post
    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"...
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    Super Member Bernhard's Avatar
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    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
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    MHF Contributor Swlabr's Avatar
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    Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups

    Quote Originally Posted by Bernhard View Post
    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)...
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    Super Member Bernhard's Avatar
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    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
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    MHF Contributor Drexel28's Avatar
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    Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups

    Quote Originally Posted by Bernhard View Post
    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}).
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    Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups

    Quote Originally Posted by Drexel28 View Post
    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?
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    MHF Contributor Drexel28's Avatar
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    Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups

    Quote Originally Posted by Deveno View Post
    do you mean GL:Vect-->Grp is a functor?
    You bet I do.
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