# General Linear Group - Problem from Dummit and Foote

• 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 $S_F$.