Heisenberg group in characteristic two

Question

I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \times V \to k$, one of these groups is $V \times k$ with the operation $(v,a) \cdot (w,b) = (v+w,a+b+\frac12 \omega(v,w))$. The other group is the subgroup of $\text{GL}_{n+2}(k)$ consisting of matrices with $1$s along the diagonal and $0$s elsewhere, except for the top row and rightmost column. This construction has the advantage of working over $k$ of arbitrary characteristic, but unfortunately uses coordinates.

I have two questions:

1. When $k$ does not have characteristic $2$, are these two groups isomorphic?

2. Is there a coordinate-free construction of the Heisenberg group in characteristic $2$ (preferably in terms of a symplectic vector space over $k$)?

Edit: Will Sawin has answered the first question in the affirmative. The second question still remains, so let me phrase it more precisely: if $k$ is a field of characteristic $2$ and $V$ a symplectic vector space over $k$, can we give a coordinate-free construction of the Heisenberg group in terms of $V$ which is isomorphic to the matrix group described above? (I am not even sure that we can always choose a symplectic basis in characteristic $2$.)

Answer 1

Part 1:

To show that this is isomorphic to the other one, fix a symplectic basis for the symplectic vector space. The isomorphism sets the coefficients of the basis elements of $v$ equal to the matrix entires in the top row and rightmost column other than the top-right corner, and $a$ equal to the entry in the top-right corner minus an adjustment which is the product of the top row and rightmost column of the matrix divided by two. For simplicity, look at three-by-three matrices:

$$\begin{pmatrix} 1 & x_1 & z_1 \\ 0 & 1 & y_1 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & x_2 & z_2 \\ 0 & 1 & y_2 \\ 0 & 0 & 1 \end{pmatrix}=\begin{pmatrix} 1 & x_1+x_2 & z_1+z_2+x_1y_2 \\ 0 & 1 & y_1+y_2 \\ 0 & 0 & 1 \end{pmatrix} $$

Set $a=z+xy/2$, then:

$a_{12}=z_1+z_2+x_1y_2-(x_1+x_2)(y_1+y_2)/2=a_1+a_2+(x_1y_2-x_2y_1)/2$

This is the formula for the first definition.

This gives the isomorphism.

Part 2:

One cannot solve this problem, using just a symplectic structure, in characteristic $2$. The problem is, given a symplectic vector space, to canonically form a group, isomorphic to the Heisenberg group, with a surjection onto the vector space. A two-dimensional vector space over $\mathbb F_2$ has a unique symplectic structure, but there are three different surjections from the Heisenberg group $D_4$ to it.

I guess one could solve it by placing some additional structure on the space. I'm not sure what that structure should be.