# Groups with isomorphic factors OA-2-H3-to-H5

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

H3: Let \(k=\mathbb F_2\), \(R=k[t]\), \(A=R^3\), and \(V=k^4\). All tensor products are over \(k\). Put
\[
C=(A\otimes_k A)^{\rm Flip},\quad \Delta(a)=a\otimes a,\quad
D=(A\otimes_k V^*)\oplus C,
\]
where \({\rm Flip}(a\otimes b)=b\otimes a\), and regard the vector spaces as discrete additive groups.
The group \({\rm SL}_3(R)\) acts on \(A\) by its usual action, on \(C\) by the tensor action, and trivially on \(V^*\).

Let \(\widehat D={\rm Hom}_k(D,k)\) with the compact product topology, identified with the character group by \(\chi\mapsto(d\mapsto(-1)^{\chi(d)})\). The \({\rm SL}_3(R)\)-action is \(l\chi=\chi\circ l^{-1}\), where \(l(u,c)=(lu,lc)\).
For any basis \(\varphi_1,\ldots,\varphi_4\) of \(V^*\), set
\[
d_j=(e_1\otimes\varphi_j,0)\quad(1\le j\le4),\qquad
d_5=(0,\Delta(e_1)).
\]
For every \({\rm SL}_3(R)\)-invariant Borel probability measure \(\mu\) on \(\widehat D\),
\[
\mu(\widehat D\setminus\{0\})\le12\sum_{j=1}^5\mu\{\chi:\chi(d_j)=1\}.
\]

Prove H5: Let \(k=\mathbb F_2\), \(R=k[t]\), \(A=R^3\), and \(C=(A\otimes_k A)^{\rm Flip}\), where \({\rm Flip}(a\otimes b)=b\otimes a\). All tensor products are over \(k\), and the vector spaces are regarded as discrete additive groups.
On the symmetric tensor basis associated with \(\mathcal B=\{t^ne_j:n\ge0,\ 1\le j\le3\}\), define the \(k\)-linear map \(\delta:C\to A\) by
\[
\delta(b\otimes b)=b,\qquad
\delta(b\otimes b'+b'\otimes b)=0\quad(b\ne b').
\]
Let \(V=k^4\), with coordinate basis \(v_1,\ldots,v_4\), and set
\[
\omega(v,w)=\sum_{j=1}^2(a_jb'_j+b_ja'_j),\qquad
r_0(v)=\sum_{j=1}^2a_jb_j,\qquad Q={\rm Sp}(V,\omega),
\]
where \(v=(a_1,b_1,a_2,b_2)\), \(w=(a'_1,b'_1,a'_2,b'_2)\).
For \(q\in Q\), define \(\ell_q\in V^*\) by
\[
\ell_q\left(\sum_{j=1}^4x_jv_j\right)=\sum_{j=1}^4x_jr_0(q^{-1}v_j).
\]
Put \(D=(A\otimes_k V^*)\oplus C\), \(H={\rm SL}_3(R)\times Q\), and, for \(i=1,2\),
\[
\theta_i(l,q)(u,c)=\bigl((l,q)u+(i-1)l\delta(c)\otimes\ell_q,\,lc\bigr).
\]
Here \(lc=(l\otimes l)c\), \((l,q)(a\otimes\varphi)=la\otimes(q\varphi)\), and \((q\varphi)(v)=\varphi(q^{-1}v)\).
Define \(\Gamma_i\) by generators \(d\in D\), \(h\in H\), the additive relations of \(D\), the multiplication relations of \(H\), and \(hdh^{-1}=\theta_i(h)d\).

The groups are not isomorphic as abstract groups:
\[
\Gamma_1\not\cong\Gamma_2.
\]

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.