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Math Help - General Linear Group - Problem from Dummit and Foote

  1. #1
    Super Member Bernhard's Avatar
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    General Linear Group - Problem from Dummit and Foote

    Can anyone help with the following problem from Dummit and Foote Section 1.4 Matrix Groups.

    Show that GL_n(F) is a finite group if and only if F has a finite number of elements.

    I cannot figure out how to compose & write the proof but suspect that F being finite means a finite number of possible matrices ... hence GL_n(F) is finite

    Is it as simple as that (in principle anyway)?

    Peter
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: General Linear Group - Problem from Dummit and Foote

    Quote Originally Posted by Bernhard View Post
    Can anyone help with the following problem from Dummit and Foote Section 1.4 Matrix Groups.

    Show that GL_n(F) is a finite group if and only if F has a finite number of elements.

    I cannot figure out how to compose & write the proof but suspect that F being finite means a finite number of possible matrices ... hence GL_n(F) is finite

    Is it as simple as that (in principle anyway)?

    Peter
    Yes, that is one direction. If F is finite then there are only |F|*n^2 possible matrices (including those of determinant 0), as there are |f| choices for each position in a given matrix.

    To prove that if F is an infinite field then GL_n(F) is infinite, think about the matrix kI, where I is the n\times n identity matrix and k\in F\setminus\{0\}.
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    MHF Contributor Drexel28's Avatar
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    Re: General Linear Group - Problem from Dummit and Foote

    Another possibility for the one direction is to realize that \text{GL}_n(F) sits nicely inside S_F.
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