General Linear Group - Problem from Dummit and Foote

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October 11th 2011, 03:10 AM
Bernhard

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
October 11th 2011, 04:03 AM
Swlabr

Re: General Linear Group - Problem from Dummit and Foote

Quote:

Originally Posted by Bernhard

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 $k\in F\setminus\{0\}$ .
October 11th 2011, 09:07 AM
Drexel28

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 $S_F$ .