Thread: 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 (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 (F) is finite

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

Peter

Originally Posted by Bernhard

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

Show that (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 (F) is finite

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

Peter

Yes, that is one direction. If is finite then there are only possible matrices (including those of determinant 0), as there are choices for each position in a given matrix.

To prove that if is an infinite field then is infinite, think about the matrix , where is the identity matrix and .

Another possibility for the one direction is to realize that sits nicely inside .