We will construct the exceptional isomorphism .

The group acts on preserving the usual pairing where the are the usual basis vectors.

There is an invariant line , the span of and an invariant hyperplane . Let . acts on .

Since is the radical of the pairing on , the pairing descends to a non-degenerate bilinear pairing on . As it is symmetric and we are in characteristic 2, it is automatically skew-symmetric.

The -action preserves this pairing, hence we get our desired homomorphism from to .

To check injectivity, it suffices to show that is not in the kernel, since we know all normal subgroups of . Surjectivity then follows by a counting argument, so we get our desired isomorphism.

I’m not sure why the tex looks so bad here (this blog is self-hosted which probably contributes to the issue). If anyone can help out, please reach out to me, preferably via email.

This is an old comment and the tex has been fixed using http://www.petermc.net/blog/2019/02/08/latex-in-wordpress/