# Groups with isomorphic factors OA-2-H1-to-H2

## 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

H1: 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 \(hdh^{-1}=\theta_i(h)d\).

Each \(\theta_i\) is an action of \(H\) by additive automorphisms of \(D\), and the canonical maps from \(D\) and \(H\) give an isomorphism
\[
D\rtimes_{\theta_i}H\cong\Gamma_i\qquad(i=1,2).
\]

Prove H2: 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 \(hdh^{-1}=\theta_i(h)d\).

There is a trace-preserving normal \(*\)-isomorphism
\[
L(\Gamma_1)\cong L(\Gamma_2).
\]
For a discrete group \(\Gamma\), \(L(\Gamma)\) is generated by its left regular representation on \(\ell^2(\Gamma)\), with canonical trace \(\tau(\lambda_g)=\mathbf1_{g=1}\).

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.