If V has finite dimension does this imply that GL(V) is finite?
If $\displaystyle V$ is a finite dimensional vector space over the field $\displaystyle F$ say of degree $\displaystyle n$. Then there is an vector space isomorphism $\displaystyle \theta : V\to F^n$. Therefore, $\displaystyle \text{GL}(V)$ is isomorphic to $\displaystyle \text{GL}(n,F)$ which is a finite group.