General Linear Group - Problem from Dummit and Foote

• Oct 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 $\displaystyle GL_n$(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 $\displaystyle GL_n$(F) is finite

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

Peter
• Oct 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 $\displaystyle GL_n$(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 $\displaystyle GL_n$(F) is finite

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

Peter

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

To prove that if $\displaystyle F$ is an infinite field then $\displaystyle GL_n(F)$ is infinite, think about the matrix $\displaystyle kI$, where $\displaystyle I$ is the $\displaystyle n\times n$ identity matrix and $\displaystyle k\in F\setminus\{0\}$.
• Oct 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 $\displaystyle \text{GL}_n(F)$ sits nicely inside $\displaystyle S_F$.