General linear group inclusion

Question

Do we have $\operatorname{GL}(n,F)\le \operatorname{O}(2n,F)$ where O means general orthogonal group and $F$ is an algebraically closed field? I checked some finite group cases: $\operatorname{GL}(2,5)\le \operatorname{GOPlus}(4,5)$ and $\operatorname{GL}(3,5)\le \operatorname{GOPlus}(6,5)$. The inclusion was true.

Answer 1

Yes. Here is an abstract way to see this. If $V$ is a finite-dimensional vector space over $F$, we equip the direct sum $V \oplus V^{\ast}$ with the bilinear form

$$B(v_1 \oplus f_1, v_2 \oplus f_2) = f_1(v_2) + f_2(v_1).$$

This is a nondegenerate symmetric bilinear form, and since it is defined using only natural operations it is preserved by the action of $GL(V)$, so we get an embedding $GL(V) \hookrightarrow O(V \oplus V^{\ast}, B)$.

Over a field containing $i$ (and having characteristic $\neq 2$), which includes $\mathbb{F}_5$, this form is equivalent to the standard symmetric bilinear form so we get the desired result. In general (again assuming the characteristic $\neq 2$) it's equivalent to the diagonal form with signature $(\dim V, \dim V)$.