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Math Help - Finite group GL2(Z/nZ). Need help on contrapositive

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    Forum Admin topsquark's Avatar
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    Finite group GL2(Z/nZ). Need help on contrapositive

    Let F be a field. Show that if GL_n(F) is finite then F is also finite.
    I don't really have any good thoughts on how to start this, so I've been looking at a possible contradiction proof or the contrapositive statement, which is all I'm asking about for now. Is the contrapositive statement: "Show that if F is not finite then GL_n(F) is not finite"? The reason that I'm not sure is that it seems vastly easier to prove than the original statement, which is something I really didn't expect. I expected them to be about the same difficulty.

    Thanks!
    -Dan
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    Re: Finite group GL2(Z/nZ). Need help on contrapositive

    Hi,
    By GLn(F) I assume you mean the group of non-singular n by n matrices with entries from F. Let x in F be non-zero and A in GLn(F) the diagonal matrix with A(1,1)=x and all other entries on the diagonal 1. This establishes a one to one correspondence between non-zero elements of F and a subset of GLn(F). Clearly then F is finite provided GLn(F) is finite.
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    Re: Finite group GL2(Z/nZ). Need help on contrapositive

    Quote Originally Posted by johng View Post
    Hi,
    By GLn(F) I assume you mean the group of non-singular n by n matrices with entries from F. Let x in F be non-zero and A in GLn(F) the diagonal matrix with A(1,1)=x and all other entries on the diagonal 1. This establishes a one to one correspondence between non-zero elements of F and a subset of GLn(F). Clearly then F is finite provided GLn(F) is finite.
    Thank you for the solution. However I was simply asking if I got the contrapositive statement correct. Logic is not one of my stronger attributes.

    -Dan
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    Re: Finite group GL2(Z/nZ). Need help on contrapositive

    Yes, the contrapositive is correct.
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