Here GL(n,K) is the general linear group of degree n over a field K.
Show that GL(n,K) is a finite group if and only if K is a finite field.
$\displaystyle k$ a field and $\displaystyle n$ a positive integer.
$\displaystyle \forall x\in k,\ x.I_n\in GL(n,k)\Rightarrow (GL(n,k)\ \text{finite}\Rightarrow k\ \text{finite})$
$\displaystyle k\ \text{finite}\Rightarrow k^{n\times n} \text{finite}\Rightarrow GL(n,k)\ \text{finite}$