# Subgroup averages and character twists

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

The embedded subgroups \(K_v\cong\mathbb F_\ell^2\) of the glued group have averages \(P_v\), idempotents in the group algebra. Weighted by the matrices \(x_vx_v^{\mathsf T}\), these averages produce a rectangular matrix identity \(UT=I_m\): an \(m\times t\) matrix with a right inverse, with \(t>m\). A rectangular one-sided inverse is not yet a counterexample; to make the matrices square, the \(t\) intermediate coordinates are packed into \(m\) slots, using idempotents attached to different characters of a cyclic group. This lesson builds the group in which these characters can be changed by conjugation, and proves the identities that the packing in [Packing and the reverse defect](packing-and-the-reverse-defect.md) uses.

- Averages over the local groups (Section 1).
- The rectangular identity (Section 2).
- A twisted extension (Section 3).
- Character idempotents (Section 4).

Throughout, \(\ell\), \(K\), \(\zeta\), \(\kappa=\ell^2\), \((V,\mathcal F)\), \(t\), \(m\) and the vectors \(x_v\) are as in Theorem 1.1 of [The glued group and planar balance](the-glued-group-and-planar-balance.md), and \(N\), \(K_v\), \(L_f\) are the glued group and its subgroups from Section 4 there. The integers \(\ell\) and \(\kappa\) are odd, so they are invertible in \(K\); in characteristic two they are even equal to \(1\), but we keep them visible.

## 1. Averages over the local groups

For \(v\in V\) put
\[
P_v=\kappa^{-1}\sum_{u\in K_v}u\in K[N] .
\]

**Lemma 1.1.** \(P_v^2=P_v\), and \(uP_v=P_vu=P_v\) for \(u\in K_v\).

*Proof.* Multiplication by \(u\in K_v\) permutes \(K_v\), so \(u\sum_{w\in K_v}w=\sum_{w\in K_v}w\), and the same on the right. Hence \(P_v^2=\kappa^{-1}\sum_{u\in K_v}uP_v=\kappa^{-1}\kappa P_v=P_v\). \(\square\)

For a line \(f\) write \(\sigma_f=\sum_{u\in L_f}u\in K[N]\). This element is the same for every point \(v\in f\), because \(L_f\) is one subgroup of \(N\).

**Lemma 1.2.** For every point \(v\),
\[
\sum_{u\in K_v}u=1+\sum_{f\ni v}\bigl(\sigma_f-1\bigr) .
\]

*Proof.* The \(\ell+1\) subgroups \(L_f\), \(f\ni v\), cover \(K_v\) and meet pairwise only in \(1\). So every \(u\ne1\) of \(K_v\) occurs in exactly one \(\sigma_f-1\), and \(1\) occurs only in the first term. \(\square\)

The identity takes place inside \(K_v\); nothing about intersections of different \(K_v\) is used.

## 2. The rectangular identity

Write \(W_v=x_vx_v^{\mathsf T}\), an \(m\times m\) matrix over \(K\), and fix an ordering of \(V\). Define matrices over \(K[N]\):
\[
\begin{gathered}
U\in\operatorname{Mat}_{m\times t}(K[N]),\\
\text{column }v\text{ of }U=-\kappa\,x_vP_v,\\
T\in\operatorname{Mat}_{t\times m}(K[N]),\\
\text{row }v\text{ of }T=P_v\,x_v^{\mathsf T},
\end{gathered}
\]
and \(P=\operatorname{diag}_{v\in V}(P_v)\in\operatorname{Mat}_t(K[N])\). The entries of \(x_v\) are scalars and commute with every element of \(K[N]\).

**Proposition 2.1.**
\[
UP=U,\qquad PT=T,\qquad UT=I_m .
\]

*Proof.* The first two identities follow from \(P_v^2=P_v\). For the third,
\[
\begin{aligned}
UT&=\sum_{v\in V}(-\kappa\,x_vP_v)(P_vx_v^{\mathsf T})\\
&=-\kappa\sum_vW_vP_v=-\sum_vW_v\sum_{u\in K_v}u .
\end{aligned}
\]
By Lemma 1.2 this equals
\[
-\sum_vW_v-\sum_v\sum_{f\ni v}W_v(\sigma_f-1) .
\]
In the double sum each pair \(v\in f\) occurs once, and \(\sigma_f-1\) does not depend on \(v\), so it equals \(\sum_f\bigl(\sum_{v\in f}W_v\bigr)(\sigma_f-1)\), which is zero by the line identities \(\sum_{v\in f}W_v=0\). The total identity \(\sum_vW_v=-I_m\) gives \(UT=I_m\). \(\square\)

So the \(m\times t\) matrix \(U\) has a right inverse. Over a field, such a matrix with \(t>m\) has no left inverse, and the corresponding defect \(TU-P\) is what the counterexample is made of. The regrouping of the double sum is where the glued group matters: the same element \(\sigma_f\) appears at every point of \(f\).

## 3. A twisted extension

Let \(z\) generate a cyclic group of order \(\ell\), and put \(H=N\times\langle z\rangle\); the element \(z\) is central in \(H\). Choose a point \(v_*\in V\) and write \(K_*=K_{v_*}\). Each \(K_v\) carries the coordinates of \(\mathbb F_\ell^2\) transported by \(\theta_v\), and \(\varphi_v:K_v\to K_*\) denotes the isomorphism that keeps the coordinates. A *linear form* \(\alpha:K_*\to\mathbb F_\ell\) is a homomorphism to the additive group of \(\mathbb F_\ell\); there are exactly \(\kappa=\ell^2\) of them. Exponents in \(\mathbb F_\ell\) are read in \(\mathbb Z/\ell\).

Since \(t\le m\kappa\), we can assign to each point \(v\) a *slot* \(s(v)\in\{1,\ldots,m\}\) and a linear form \(\alpha_v\) so that the pairs \((s(v),\alpha_v)\) are distinct. Points in the same slot then have distinct forms.

**Definition 3.1.** \(L\) is the group generated by \(H\) and new elements \(\tau_v\), \(v\in V\), subject to the relations of \(H\) and
\[
\begin{gathered}
\tau_v\,(uz^r)\,\tau_v^{-1}=\varphi_v(u)\,z^{r+\alpha_v(\varphi_v(u))}\\
(u\in K_v,\ r\in\mathbb Z/\ell).
\end{gathered}
\]

**Lemma 3.2.** For each \(v\), the map \(\psi_v:K_v\times\langle z\rangle\to K_*\times\langle z\rangle\), \(uz^r\mapsto\varphi_v(u)z^{r+\alpha_v(\varphi_v(u))}\), is a group isomorphism.

*Proof.* Since \(z\) is central, \(\varphi_v\) is a homomorphism and \(\alpha_v\circ\varphi_v\) is additive,
\[
\begin{aligned}
&\psi_v(uz^r)\psi_v(u'z^{r'})\\
&\quad=\varphi_v(uu')\,z^{r+r'+\alpha_v(\varphi_v(uu'))}\\
&\quad=\psi_v(uu'z^{r+r'}) .
\end{aligned}
\]
The map \(wz^r\mapsto\varphi_v^{-1}(w)z^{r-\alpha_v(w)}\) is inverse to it. \(\square\)

**Lemma 3.3** ([OpenAI-DF2, Section 4.2]). The natural map \(H\to L\) is injective. Consequently \(K[H]\) is a subalgebra of \(K[L]\).

*Proof.* We build an action of \(L\) on the set \(H\) that extends the action of \(H\) on itself by left multiplication, \(\lambda(h)x=hx\). Write \(A_v=K_v\times\langle z\rangle\) and \(A_*=K_*\times\langle z\rangle\), subgroups of order \(\ell^3\). Under left multiplication by \(A_v\), the set \(H\) splits into the right cosets \(A_vh\), each a copy of \(A_v\) with its regular action; the same holds for \(A_*\). The two sets of cosets have the same cardinality: both have \(|H|/\ell^3\) elements if \(H\) is finite, and both are countably infinite otherwise, because \(H\) is finitely generated, hence countable, and the subgroups are finite. Choose a bijection between them, that is, representatives \(h_i\) and \(k_i\) with cosets \(A_vh_i\) paired with \(A_*k_i\), and define \(F_v:H\to H\) by
\[
F_v(ah_i)=\psi_v(a)\,k_i\qquad(a\in A_v).
\]
This is a bijection, and for \(b\in A_v\), \(F_v(bah_i)=\psi_v(b)\psi_v(a)k_i\), that is,
\[
F_v\,\lambda(b)=\lambda(\psi_v(b))\,F_v .
\]
Send each element of \(H\) to its left multiplication and each \(\tau_v\) to \(F_v\). The relations of \(H\) hold because \(\lambda\) is a homomorphism, and the relations of Definition 3.1 hold by the display. Different stable letters occur in different relations, so the bijections \(F_v\) can be chosen independently. This defines a homomorphism from \(L\) to the permutations of \(H\) whose restriction to \(H\) is \(\lambda\), which is injective. So \(H\to L\) is injective, and so is \(K[H]\to K[L]\). \(\square\)

Definition 3.1 adds \(|V|\) generators and finitely many relations to a presentation of \(H\). The extension is a multiple HNN extension; Lemma 3.3 proves directly the embedding property that is usually derived from the normal form for such extensions.

## 4. Character idempotents

Taking \(u=1\) and \(r=1\) in Definition 3.1 gives \(\tau_vz\tau_v^{-1}=z\). So \(z\) is central in \(L\), and so is
\[
e=\ell^{-1}\sum_{r=0}^{\ell-1}\zeta^{-r}z^r\in K[L] .
\]

**Lemma 4.1.** \(e\) is a nonzero central idempotent and \(ez=\zeta e\).

*Proof.* The coefficient of \(z^a\) in \(e^2\) is \(\ell^{-2}\sum_{r+r'=a}\zeta^{-r}\zeta^{-r'}=\ell^{-2}\cdot\ell\,\zeta^{-a}\), the coefficient of \(z^a\) in \(e\). Reindexing \(ez=\ell^{-1}\sum_r\zeta^{-r}z^{r+1}\) gives \(\zeta e\). The coefficient of the identity in \(e\) is \(\ell^{-1}\ne0\), and the elements \(z^r\) are distinct in \(L\) by Lemma 3.3. \(\square\)

For a linear form \(\alpha\) on \(K_*\) put
\[
Q_\alpha=\kappa^{-1}\sum_{u\in K_*}\zeta^{\alpha(u)}u\in K[L] .
\]

**Lemma 4.2.** \(Q_\alpha Q_\beta=Q_\alpha\) if \(\alpha=\beta\), and \(Q_\alpha Q_\beta=0\) otherwise.

*Proof.* Write the group \(K_*\) additively in its coordinates. The coefficient of \(w\) in \(Q_\alpha Q_\beta\) is
\[
\begin{aligned}
&\kappa^{-2}\sum_{u\in K_*}\zeta^{\alpha(u)+\beta(w-u)}\\
&\quad=\kappa^{-2}\zeta^{\beta(w)}\sum_{u}\zeta^{(\alpha-\beta)(u)} .
\end{aligned}
\]
If \(\alpha=\beta\), the last sum is \(\kappa\), and the coefficient is that of \(w\) in \(Q_\alpha\). Otherwise choose \(u_0\) with \(\zeta^{(\alpha-\beta)(u_0)}\ne1\); replacing \(u\) by \(u+u_0\) multiplies the sum by this number without changing it, so the sum is zero. \(\square\)

**Lemma 4.3** (twisted averages). For every point \(v\),
\[
\tau_v\,eP_v\,\tau_v^{-1}=e\,Q_{\alpha_v} .
\]

*Proof.* \(e\) is central, and conjugation by \(\tau_v\) maps \(u\in K_v\) to \(\varphi_v(u)z^{\alpha_v(\varphi_v(u))}\). So, with \(w=\varphi_v(u)\),
\[
\begin{aligned}
\tau_veP_v\tau_v^{-1}&=\kappa^{-1}\sum_{w\in K_*}e\,z^{\alpha_v(w)}w\\
&=\kappa^{-1}\sum_{w\in K_*}\zeta^{\alpha_v(w)}e\,w=eQ_{\alpha_v},
\end{aligned}
\]
using \(ez^r=\zeta^re\). \(\square\)

So conjugation by \(\tau_v\) carries the average \(eP_v\) over \(K_v\) to the character idempotent \(eQ_{\alpha_v}\) over the single group \(K_*\). Two points in the same slot have distinct forms, so their twisted averages are orthogonal by Lemma 4.2. This is what lets them share a slot in [Packing and the reverse defect](packing-and-the-reverse-defect.md).

## 5. Exercises

**5.1** (easy). Show that in characteristic two \(\kappa=1\) and \(\ell=1\) in \(K\), so that \(P_v=\sum_{u\in K_v}u\) and \(e=\sum_r\zeta^{-r}z^r\).

**5.2** (easy). Show that \(\sum_\alpha Q_\alpha=1\), the sum over all \(\kappa\) linear forms on \(K_*\).

**5.3** (medium). Show that a slot assignment with distinct pairs \((s(v),\alpha_v)\) exists exactly when \(t\le m\kappa\), and that if two points \(v\ne w\) in the same slot had the same form, then \(eQ_{\alpha_v}\cdot eQ_{\alpha_w}=eQ_{\alpha_v}\ne0\).

**5.4** (medium). Let \(\ell=3\) and \(K=\mathbb F_4\), with \(\zeta\) a root of \(\zeta^2+\zeta+1\). Write out \(e\) and check \(e^2=e\) and \(ez=\zeta e\) directly in \(K[\langle z\rangle]\).

## 6. Solutions

**5.1.** In characteristic two, an odd integer is \(1\) in \(K\), and \(\kappa=\ell^2\) and \(\ell\) are odd.

**5.2.** The coefficient of \(u\) in \(\sum_\alpha Q_\alpha\) is \(\kappa^{-1}\sum_\alpha\zeta^{\alpha(u)}\). For \(u=1\) (the zero vector) every term is \(1\), giving \(1\). For \(u\ne1\) there is a form \(\beta\) with \(\beta(u)\ne0\); replacing \(\alpha\) by \(\alpha+\beta\) multiplies the sum by \(\zeta^{\beta(u)}\ne1\) without changing it, so the sum is zero.

**5.3.** There are \(m\kappa\) pairs, so \(t\) distinct pairs exist exactly when \(t\le m\kappa\). If \(\alpha_v=\alpha_w\), then \(eQ_{\alpha_v}eQ_{\alpha_w}=eQ_{\alpha_v}^2=eQ_{\alpha_v}\), using that \(e\) is a central idempotent, and \(eQ_{\alpha_v}\ne0\) because its coefficient at the identity is \(\ell^{-1}\kappa^{-1}\ne0\) (the elements \(z^ru\) are distinct in \(H\)). The cross term would then not vanish.

**5.4.** Here \(\ell^{-1}=1\) and \(\zeta^{-1}=\zeta^2=\zeta+1\), so \(e=1+\zeta^2z+\zeta z^2\). Then
\[
e^2=1+\zeta^4z^2+\zeta^2z^4=1+\zeta z^2+\zeta^2z ,
\]
since squaring is additive in characteristic two, \(\zeta^3=1\) and \(z^3=1\); this is \(e\). Also \(ez=z+\zeta^2z^2+\zeta\) and \(\zeta e=\zeta+z+\zeta^2z^2\), which agree.

## References

- [OpenAI-DF2] OpenAI, *A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two*, OpenAI Math Release preprint, 23 September 2026, Sections 4.1 and 4.2. https://github.com/openai/math/tree/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026