Is every subgroup (under composition) of $GL(V)$ isomorphic to $GL(W)$ for some subspace $W$ of $V$?

Question

In the above, $V$ is an arbitrary finite vector space over $\mathbb{C}$.

I ask this because I am thinking of an alternative definition of the equivalence of representations.

Answer 1

Let $V = \mathbb{C}$, so that $GL(V) \cong \mathbb{C}^\times$.

Now, the only subspaces $W$ of $V$ are $V$ itself and $\{ 0 \}$. So the only possibilities for $GL(W)$ are $\mathbb{C}^\times$, or the trivial group.

But $\mathbb{C}^\times$ has subgroups not isomorphic to those (the $n$th roots of unity, for example).

So the claim is not true.