The finiteness problem.

For finite \(G\), the algebra and all its finite matrix rings are finite-dimensional over \(K\). A one-sided inverse between their left multiplication operators is therefore two-sided. This argument works even when the characteristic divides \(|G|\). The difficulty for an infinite group is to obtain a corresponding conclusion from finite support alone.

Kaplansky proved stable finiteness in characteristic zero using finite von Neumann algebras, and asked whether the one-sided-inverse property persists in positive characteristic (Kaplansky 1972, 122–23). Ara, O’Meara, and Perera proved stable finiteness for free-by-amenable groups over arbitrary division rings (Ara et al. 2002). Elek and Szabó (Elek and Szabó 2004, Corollary 4.7) established the same conclusion for all sofic groups. Such groups admit finite permutation models that approximate multiplication while keeping distinct elements separated. These theorems are constraints on the example, not hypotheses about its group. In particular, Elek–Szabó’s theorem implies, after the scalar construction is completed, that \(G\) cannot be sofic.

A separate characteristic-two construction uses incidence structures, polynomial pairings, a planar group-embedding argument, and character packing (OpenAI 2026, Theorem 1.1). The construction here has a different local mechanism. A high-degree subspace of a modular translation representation is annihilated on one side; a weighted pairing gives a compatible annihilation on the other. The characteristic-two theorem is not an input to this odd-prime argument, and the two methods retain distinct coefficient hypotheses.

The two quantities that must be compared.

A matrix \(F\) fits a simple graph when \(F_{ii}=1\) and \(F_{ij}=0\) for distinct nonadjacent vertices. We use a triangle-free graph on \(m\) vertices and a specified factorization \[F_{ij}=\sum_{T\in\mathcal T}\alpha_i(T)\beta_j(T),\qquad |\mathcal T|=t,\qquad \frac{t}{m}<\frac1{100}.\] This is an explicit threshold-Kneser instance of the polynomial constructions of Haviv and Golovnev–Haviv (Haviv 2018, Proposition 13) (Golovnev and Haviv 2022, Theorem 3.2 and Lemma 3.5), whose polynomial ancestry includes Alon–Babai–Suzuki (Alon et al. 1991). Low-rank fitting matrices also give storage-code recovery equations. The triangle-free constructions of Barg–Zémor, Huang–Xiang, and Barg–Schwartz–Yohananov provide related context (Barg and Zémor 2022; Huang and Xiang 2023; Barg et al. 2024). Their results are not used to supply a group-algebra idempotent.

At every vertex we use a copy of a finite group \(H\) and an idempotent \(e\in K[H]\). Write \(w=|H|\) and \(d=\dim_K eK[H]\), the rank of left multiplication by \(e\) on the regular representation. The second quantity is \[\frac d w>\frac1{96}.\] The idempotents satisfy a directed condition: after identifying certain cyclic subgroups at adjacent vertices, \(e_i e_j=0\) for every edge with \(i<j\). There is no assertion about the reverse product. The asymmetry is what allows a triangular calculation to combine the two strict bounds.

The characteristic-dependent construction appears in Section 3. High augmentation degree gives the annihilated columns, in the elementary abelian setting of Jennings’ theory (Jennings 1941). A diagonal weighted Gram determinant and Schwartz’s grid bound (Schwartz 1980, Lemma 1 and Corollary 1), with related polynomial-identity work of Zippel (Zippel 1979), supply enough nondegenerate pairings. Fourier character orbits turn these projections into actual elements of \(K[H]\). The full algebra need not be semisimple; only its prime-to-\(p\) normal abelian subgroup is diagonalized, in the character-orbit setting of Clifford (Clifford 1937).

From the comparison to scalar elements.

The proof has three transitions that numerical rank inequalities alone cannot justify. First, the quotient identifying the cyclic subgroups must retain the finite vertex groups. Its injectivity proof uses the \(C(4)\)–\(T(4)\) graphical-complex mechanism described by Prytuła (Prytuła 2024, sec. 8): local short-word restrictions and triangle-freeness rule out a spherical map with exactly one nontrivial face. Ordinary graph-of-groups normal form does not by itself establish this first quotient’s injectivity. Section 4 proves it locally, including the reductions for loops and bridges.

Second, regular ranks must become identities over the same ring. A further group enlargement \(G_1\) has a nonzero central idempotent \(f\) in \(R=K[G_1]\). In the ring \(S=fR\), whose identity is \(f\), the elements \(P_i=fe_i\) admit rectangular matrices \[J_i\in\operatorname{Mat}_{w\times d}(S),\qquad Y_i\in\operatorname{Mat}_{d\times w}(S),\qquad Y_iJ_i=fI_d,\quad J_iY_i=P_iI_w.\] The finite-group blocks in this step have elementary abelian \(p\)-group stabilizers, possibly nontrivial and nonnormal. Their group algebras have nilpotent augmentation ideals, and a finite change of basis proves exact integer rank identities before coordinates are paired. Section 5 gives the full construction of these matrices, with their entries in explicitly attached finite subgroups.

Third, a matrix counterexample does not yet give two scalar elements of the same group algebra. In Section 6, the fit and the ordered zero products produce a split injection \[S^{md}\longrightarrow S^{wt}\longrightarrow S^{md}, \qquad wt<md,\] whose composite is the identity. We derive square matrices with a one-sided inverse and certify the reverse defect by a nonzero kernel vector. Section 7 then implements the finite-factor reduction of Dykema–Juschenko (Dykema and Juschenko 2015, Lemma 2.1 and Theorem 2.2). A finite Heisenberg group gives explicit matrix units, their linear independence makes the encoding injective, and a complementary idempotent restores the full scalar identity. The shift and modulation model is classical (Gurevich and Hadani 2009, sec. 1.2); the composite cyclic order and finite coefficient field used here are checked directly.

The dynamical consequence.

A configuration over a finite alphabet \(A\) is any function \(G\to A\). A cellular automaton has one finite memory set \(E\subset G\) and one rule applied to \((x(gu))_{u\in E}\) at every \(g\). A group is surjunctive when every injective such automaton, over every finite alphabet, is surjective. Gottschalk proposed this for all groups (Gottschalk 1973); Gromov and Weiss proved it for sofic groups (Gromov 1999; Weiss 2000). The group-ring connection is recorded by Elek–Szabó and developed in the linear theory of Ceccherini-Silberstein–Coornaert (Elek and Szabó 2004; Ceccherini-Silberstein and Coornaert 2006). Phung proved stable finiteness over every field for surjunctive groups (Phung 2023, Theorem B); subsequent proofs and extensions include (Ceccherini-Silberstein et al. 2025a, 2025b).

Section 8 gives the elementary transfer with its actual multiplication convention. The coefficients of \(b\) define an injective automaton, those of \(a\) give a left inverse, and a nonzero coefficient of \(ba-1\) specifies a missing point indicator. The argument excludes preimages of arbitrary support. It is a consequence of the scalar construction, not a further existence theorem.

All matrices below act on columns of free right modules. Matrix entries therefore multiply coordinates on the left. Transpose is ordinary transpose over \(K\), with no involution. Cardinalities and dimensions in rank comparisons are integers, not elements of the coefficient field. For a positive integer \(n\), write \([n]=\{1,\ldots,n\}\).

The annihilated column space

On \(K^P\), let \(\mathbf e_x\) be the standard basis and \(L_s\mathbf e_x=\mathbf e_{s+x}\). For the three standard generators \(\varepsilon_j\) of \(P\) set \(X_j=L_{\varepsilon_j}-I\). The \(X_j\) commute, and characteristic \(p\) gives \(X_j^p=0\). The vectors \[X_1^{n_1}X_2^{n_2}X_3^{n_3}\mathbf e_0, \qquad 0\le n_1,n_2,n_3<p,\] are a basis. Indeed, every translation of \(\mathbf e_0\) is a product of powers of \(I+X_j\) applied to \(\mathbf e_0\), so the displayed family spans \(K^P\); its cardinality is the dimension \(p^3\).

This is the elementary abelian augmentation filtration in the setting studied by Jennings (Jennings 1941). Here the exact degree argument can be given directly. Let \(B\) have as columns the displayed basis vectors with \(n_1+n_2+n_3\ge D-\ell+1\), in order. For every \(s\), the matrix \(L_s-I\) is a polynomial in the \(X_j\) with zero constant term. Multiplication by its \(\ell\)th power raises every column’s total degree to at least \(D+1\). Such a monomial has an exponent at least \(p\), so vanishes. Applying the same assertion to \(-s\) and transposing yields \[ (L_s-I)^\ell B=0,\qquad B^{\mathsf T}(L_s-I)^\ell=0. \tag{1}\] The transpose identity uses \(L_{-s}^{\mathsf T}=L_s\); no scalar conjugation is present.

The columns are independent. Their number \(h\) can be counted by setting \(a_j=p-1-n_j\): the new nonnegative indices have sum at most \(\ell-1\). Their individual upper bounds are automatic because \(\ell-1<p\). Stars and bars gives \[ h=\binom{\ell+2}{3},\qquad \frac{h}{p^3}=\frac{(p-1)(p+1)(p+3)}{48p^3}>\frac1{48}. \tag{2}\] For the strict inequality, the numerator minus \(p^3\) is \(3p^2-p-3>0\) for \(p\ge3\).

The same subspace is killed on the two sides in Equation (1), but an arbitrary projection onto it need not give the outgoing conjugate annihilation. We therefore choose the row factor by a diagonal weighted pairing.

Many nonsingular diagonal pairings

For a finite function group \(A_*=(\mathbb Z/M\mathbb Z)^E\), its characters \(\chi:A_*\to K^*\) are all exponent lists of powers of \(\zeta\). Put \[ p_\chi^{A_*}=|A_*|^{-1}\sum_{a\in A_*}\chi(a)^{-1}a. \tag{3}\] The denominator is invertible. Reindexing the sum gives \(a p_\chi^{A_*}=\chi(a)p_\chi^{A_*}\). A nontrivial character has sum zero, since translating its sum by an element on which it is nontrivial multiplies it by a scalar different from one. It follows that \[p_\chi^{A_*}p_\psi^{A_*}=\delta_{\chi\psi}p_\chi^{A_*},\qquad \sum_\chi p_\chi^{A_*}=1.\] For the last identity, sum coefficients over the independent character exponents: every nonzero coordinate of a nonidentity element is detected by a power of \(\zeta\). Each idempotent is nonzero, there are \(|A_*|\) of them, and orthogonality makes them independent. They form a basis of \(K[A_*]\).

Specialize to \(A_0\). Its character action is \((s\lambda)(a)=\lambda((-s)\cdot a)\). Define \[\chi_x=x\lambda,\qquad D_\lambda=\operatorname{diag}(\chi_x(c):x\in P),\qquad N_\lambda=B^{\mathsf T}D_\lambda^{-1}B.\] Call \(\lambda\) admissible if its stabilizer in \(P\) is trivial and \(N_\lambda\) is invertible. This condition is a finite test.

Select the first admissible character from each \(P\)-orbit containing one, and call the set of selected characters \(\mathscr R\). Each selected orbit is free, of size \(p^3\). These orbits cover all admissible characters, so \[ |\mathscr R|>\frac{|A_0|}{2p^3}. \tag{5}\] This argument does not need admissibility to be constant on an orbit.

Realizing the projections in the group algebra

Fix \(\lambda\in\mathscr R\). For \(x,y\in P\), define \[E_{xy}^{\lambda}=p_{x\lambda}^{A_0}u(x-y).\] Conjugation gives \(u(s)p_\chi^{A_0}u(s)^{-1}=p_{s\chi}^{A_0}\). Fourier orthogonality and freeness of the orbit therefore imply \[E_{xy}^{\lambda}E_{zt}^{\lambda}=\delta_{yz}E_{xt}^{\lambda}.\] Indeed the Fourier product can survive only when \(x\lambda=(x-y+z)\lambda\), which is equivalent to \(y=z\). Products between different orbits vanish. The sum \(q_\lambda=\sum_x p_{x\lambda}^{A_0}\) is central: it commutes with \(A_0\) and its summands are permuted by \(P\). The basis \(p_{x\lambda}^{A_0}u(s)\) of its ideal is indexed equally by \((x,y)\) through \(s=x-y\). Thus these units prove \[q_\lambda K[H]\cong\operatorname{Mat}_{p^3}(K).\] This is an explicit character-orbit block, in the normal-subgroup representation setting of Clifford (Clifford 1937). It does not assert semisimplicity of the entire modular algebra.

On this block, the elements \(u(s)\), \(c\), and \(v(s)\) act respectively by \[L_s,\qquad D_\lambda,\qquad D_\lambda L_sD_\lambda^{-1}.\] For example, left multiplication by \(u(s)\) shifts row \(x\) to row \(s+x\), whereas \(c\) multiplies row \(x\) by \((x\lambda)(c)\).

Define a matrix and then a finite group-algebra element by \[ T_\lambda=BN_\lambda^{-1}B^{\mathsf T}D_\lambda^{-1},\qquad e=\sum_{\lambda\in\mathscr R}\sum_{x,y\in P} (T_\lambda)_{xy}E_{xy}^{\lambda}. \tag{6}\] The inverse exists by the selection criterion. Since \(B^{\mathsf T}D_\lambda^{-1}B=N_\lambda\), we have \(T_\lambda B=B\) and \(T_\lambda^2=T_\lambda\). Its image is exactly \(\operatorname{im}B\), of dimension \(h\). Orthogonality of the blocks proves \(e^2=e\); on the complementary ideal its component is zero.

We now compute the regular rank, rather than the smaller rank of a single representation matrix. In \(\operatorname{Mat}_{p^3}(K)\), left multiplication by \(T_\lambda\) has image the matrices whose \(p^3\) columns all lie in \(\operatorname{im}B\). Its image dimension is \(hp^3\). Summing the selected blocks gives \[ d=hp^3|\mathscr R|,\qquad \frac d w=\frac{h|\mathscr R|}{|A_0|} >\frac{h}{2p^3}>\frac1{96}. \tag{7}\]

The incoming annihilation follows from \((L_s-I)^\ell T_\lambda=0\). For the outgoing annihilation, keep the factors in their actual order: \[\begin{split} B^{\mathsf T}D_\lambda^{-1} (D_\lambda L_sD_\lambda^{-1}-I)^\ell &=B^{\mathsf T}(L_s-I)^\ell D_\lambda^{-1}\\ &=0. \end{split}\] Equation (1) proves both assertions on every selected block. They also hold on the complementary ideal, where \(e\) is zero. This proves Proposition 4, including its finite prescription and the strict rank estimate.

From opposite annihilations to an ordered zero product

Take copies \(H_i\) of \(H\). For each edge \(i<j\), identify \(v(1,\bar j,\bar j^2)\) in \(H_i\) with \(u(1,\bar i,\bar i^2)\) in \(H_j\), where bars denote residues in \(\mathbb F_p\). Let \(G_0\) be the resulting free-product quotient, and let \(e_i\) denote the image of \(e\) from \(H_i\). The injectivity of these copies is proved in Section 4. The following ordered identity already follows from the local calculations and the quotient relations.

We have constructed the large regular rank and the edge vanishings. To compare them with the graph factorization, the remaining structural step must put exact rectangular realizations of all those ranks inside one nonzero ring.

An injectivity criterion

Application to the modular construction

Recall from Section 2 the group \(H=A_0\rtimes P\), with \(P=(\mathbb F_p)^3\) and \(A_0=(\mathbb Z/M\mathbb Z)^P\). Its subgroups \(U=u(P)\) and \(V=cUc^{-1}\) satisfy \(U\cap V=\{1\}\). The quotient \(G_0\) identifies, for every edge \(i<j\), the elements \[ v(1,\bar j,\bar j^{2})\in H_i, \qquad u(1,\bar i,\bar i^{2})\in H_j, \tag{15}\] where \(\bar i\) denotes the residue of \(i\) in \(\mathbb F_p\). Both elements have order \(p\), so the corresponding maps from \(C_p\) are embeddings. We now verify the remaining local condition.

Attaching the finite groups

Let \(z\) initially generate an independent copy of \(C_M\), and write \(Q_{\mathrm{base}}=\langle z\rangle\times G_0\). For each \(i\), define \[B_i=(C_M)^{H_i},\qquad L_i=B_i\rtimes H_i, \qquad (h\cdot b)(x)=b(h^{-1}x).\] The groups \(B_i\) are written additively. Let \(\delta_i\in B_i\) be the constant function of value \(1\in C_M\). It has order \(M\) and is fixed by \(H_i\), so it is central in \(L_i\). Moreover, \(\langle\delta_i\rangle\cap H_i=1\). Thus \(C_i=\langle z\rangle\times H_i\) embeds in both \(Q_{\mathrm{base}}\) and \(L_i\), with \((z^k,h)\) sent to \((k\delta_i,h)\) in the latter. Define \(G_1\) to be the quotient of the free product \[Q_{\mathrm{base}}*L_1*\cdots*L_m\] that identifies these two copies of \(C_i\) for every \(i\).

For completeness, the following action proves that all the factors embed in this quotient. Let \(X\) be a countable disjoint union of copies of the left regular \(Q_{\mathrm{base}}\)-set. Its restriction to each \(C_i\) is free and has countably infinitely many orbits. A countable disjoint union \(X_i\) of regular \(L_i\)-sets has the same property when restricted to \(C_i\). Pair their \(C_i\)-orbits and choose one point in each paired orbit; extending those point pairings equivariantly gives a \(C_i\)-equivariant bijection \(X\to X_i\). Transport the regular \(L_i\)-action to \(X\) using this bijection. The resulting actions of \(Q_{\mathrm{base}},L_1,\ldots,L_m\) agree on every prescribed \(C_i\), hence define an action of \(G_1\) on \(X\). Each factor acts faithfully, which proves its injectivity in \(G_1\). This argument uses the assumed embeddings \(H_i\hookrightarrow G_0\) in defining the subgroups \(C_i\le Q_{\mathrm{base}}\).

The element \(z\) is central in every generating factor of \(G_1\) and therefore in \(G_1\); its order is still \(M\) because \(Q_{\mathrm{base}}\) embeds. In \(R\), direct multiplication gives \[zf=\zeta f,\qquad f^2=\frac1M\sum_{k=0}^{M-1}\zeta^{-k}z^kf=f.\] The coefficient of the identity in \(f\) is \(M^{-1}\ne0\), since the elements \(1,z,\ldots,z^{M-1}\) are distinct. Thus \(f\) is a nonzero central idempotent, and \(S=fR\) is a ring with identity \(f\). Each \(P_i\) is an idempotent in \(fK[L_i]\). Also, any identity \(e_ie_j=0\) already valid in \(K[G_0]\) remains valid after multiplication by \(f\) in \(S\).

The matrix blocks of the central sector

Fix \(i\) for the moment. A character of \(B_i\) means a homomorphism \(\chi:B_i\to K^*\). If \(b_x\) denotes the coordinate function with value \(1\) at \(x\in H_i\) and \(0\) elsewhere, then every character is uniquely specified by exponents \(a_x\in\mathbb Z/M\mathbb Z\) through \(\chi(b_x)=\zeta^{a_x}\). Define \[p_\chi=\frac1{|B_i|}\sum_{b\in B_i}\chi(b)^{-1}b \quad\text{in }K[B_i].\] Apply the Fourier calculation of Equation (3) to the finite index set \(H_i\). The denominator is invertible because \(p\nmid M\). The elements \(p_\chi\) are a basis of pairwise orthogonal idempotents whose sum is \(1\), and \(bp_\chi=\chi(b)p_\chi\).

Use the action \[(h\chi)(b)=\chi(h^{-1}\cdot b).\] Then \(hp_\chi h^{-1}=p_{h\chi}\). Since \(z=\delta_i\) is fixed by \(H_i\), the set \[\mathcal X_i=\{\chi:\chi(z)=\zeta\}\] is \(H_i\)-invariant, and the Fourier formulas give \[ f=\sum_{\chi\in\mathcal X_i}p_\chi. \tag{17}\] For each orbit in \(\mathcal X_i\), choose one representative \(\theta\) and put \(T_\theta=\operatorname{Stab}_{H_i}(\theta)\).

Eliminating the prime-to-\(p\) part of each stabilizer is what makes its augmentation ideal nilpotent, and will permit the later change of basis. We next give the block isomorphism, including its inverse. This is the character-orbit construction already used in Section 3, now allowing nontrivial stabilizers; compare the Clifford-theoretic orbit picture in (Clifford 1937). It will identify the underlying \(K\)-linear rank of \(P_i\) in every block with the same original integer \(d\). For one orbit representative \(\theta\), abbreviate \[T=T_\theta,\qquad s=[H_i:T],\qquad D=K[T].\] Choose representatives \(r_1,\ldots,r_s\) of the left cosets \(hT\). The space \(K[H_i]\) is a free right \(D\)-module with this ordered basis. Let \(g\in H_i\) act on it by left multiplication \(L_g\), and let \(b\in B_i\) act diagonally by \[D_b(h)=(h\theta)(b)h\qquad(h\in H_i).\] Both actions commute with the right action of \(T\). For the diagonal action this follows from \((ht)\theta=h\theta\) for \(t\in T\). The semidirect-product relation is also respected, because \[L_gD_bL_g^{-1}(h) =((g^{-1}h)\theta)(b)h =(h\theta)(g\cdot b)h =D_{g\cdot b}(h).\] Thus we have an algebra homomorphism \(\rho_\theta:K[L_i]\to\operatorname{Mat}_s(D)\). The element \(z\) acts as \(\zeta\), so \(\rho_\theta(f)=I_s\).

For \(t_0\in T\) and \(1\le a,b\le s\), consider \[ p_{r_a\theta}\,r_at_0r_b^{-1}. \tag{18}\] It belongs to \(fK[L_i]\) by (17). In the \(\theta\)-representation it sends \(r_bt'\) to \(r_at_0t'\) for every \(t'\in T\) and vanishes on every other basis coset. To check the last assertion, its left group factor sends the coset \(r_cT\) to \(r_at_0r_b^{-1}r_cT\); the Fourier projection keeps this coset precisely when it equals \(r_aT\), which is equivalent to \(c=b\). In a representation associated to another orbit, the Fourier projection vanishes identically. Consequently (18) maps to the matrix with entry \(t_0\) in position \((a,b)\) of the \(\theta\)-block and zero elsewhere.

Taking the product of these representations gives a surjective map \[ \rho:fK[L_i]\longrightarrow \prod_{\theta}\operatorname{Mat}_{[H_i:T_\theta]}(K[T_\theta]). \tag{19}\] It is injective as well. The elements \(p_\chi h\), for \(\chi\in\mathcal X_i\) and \(h\in H_i\), form a basis of its source. An orbit of size \(s\) contributes \(sw=s^2|T|\) basis elements, equal to the dimension of the corresponding target block. Hence the source and target have equal finite \(K\)-dimension. Formula (18), extended linearly in \(t_0\) and summed over the positions and blocks, is an explicit inverse of \(\rho\).

Turning dimensions into rectangular identities

Continue in one block and let \(E=\rho_\theta(P_i)\in \operatorname{Mat}_s(D)\). As a \(K\)-linear operator on \(K[H_i]\), \(E\) is exactly left multiplication by \(e_i\), since \(f\) acts as the identity. Therefore \[ \dim_K\operatorname{im}E=d. \tag{20}\] This is the link between the original rank calculation and the block decomposition. We will now obtain a basis in which \(E\) is a coordinate projection over \(D\), not just over \(K\). The argument is a constructive nilpotent-ideal instance of the freeness of projective modules over local rings, studied by Kaplansky (Kaplansky 1958, Theorem 2); no general projective-module theorem is needed here.

Let \(\varepsilon:D\to K\) be augmentation, and set \(\mathfrak a=\ker\varepsilon\). By Lemma 10, we may choose generators \(t_1,\ldots,t_q\) of \(T\cong(C_p)^q\) with \(0\le q\le3\). The elements \(X_j=t_j-1\) commute, satisfy \(X_j^p=0\), and generate \(\mathfrak a\). Expanding \(t_j=1+X_j\) and counting dimensions gives \[D\cong K[X_1,\ldots,X_q]/(X_1^p,\ldots,X_q^p).\] It follows that \[\mathfrak a^{D_{\max}+1}=0,\qquad D_{\max}=3(p-1).\] Indeed, a monomial of total degree greater than \(q(p-1)\) has an exponent at least \(p\). This also covers \(q=0\), when \(D=K\) and \(\mathfrak a=0\). Matrices with entries in \(\mathfrak a\) have the same nilpotence bound under multiplication, because each entry of a product of \(D_{\max}+1\) such matrices lies in \(\mathfrak a^{D_{\max}+1}\).

Put \(\overline E=\varepsilon(E)\), applying augmentation entrywise, and let \(d_0=\operatorname{rank}_K\overline E\). As \(\overline E\) is idempotent, its image and the image of \(I_s-\overline E\) form complementary subspaces of \(K^s\). Scan the columns of \(\overline E\) in order and keep each column that increases their span; then do the same for \(I_s-\overline E\). Build \(Q\) by taking the corresponding \(d_0\) columns of \(E\), followed by the corresponding \(s-d_0\) columns of \(I_s-E\). Its augmentation \(Q_0\) is invertible over \(K\). Regard \(Q_0\) as a scalar matrix over \(D\), and set \(N=Q_0^{-1}(Q-Q_0)\). Its entries belong to \(\mathfrak a\), so \(N^{D_{\max}+1}=0\). Since \(Q=Q_0(I_s+N)\), \[ Q^{-1}=\left(\sum_{k=0}^{D_{\max}}(-N)^k\right)Q_0^{-1}. \tag{21}\] Each of the first \(d_0\) columns of \(Q\) is fixed by \(E\), and every remaining column is killed by \(E\). Thus, for \(\Pi=\operatorname{diag}(I_{d_0},0)\), \[EQ=Q\Pi,\qquad Q^{-1}EQ=\Pi.\] Conjugation by \(Q\) preserves \(K\)-dimension of images. Comparing with (20) gives the integer equalities \[ d=d_0|T|,\qquad w=s|T|,\qquad wd_0=sd. \tag{22}\] In particular, these are equalities of ordinary dimensions, not equalities after reduction to \(K\).

We can now match coordinates exactly. Order pairs lexicographically and choose the increasing bijection between \[\{(\mu,l):1\le\mu\le d,\ 1\le l\le s\} \quad\text{and}\quad \{(\lambda,k):1\le\lambda\le w,\ 1\le k\le d_0\};\] the sets have equal size by (22). Let \(V\) be the \(w\)-by-\(d\) block matrix with \(s\)-by-\(s\) entries that sends coordinate \((\mu,l)\) to its paired coordinate \((\lambda,k)\). Thus \(V_{\lambda\mu}\) is the sum of standard matrix units \(E_{kl}^{\mathrm{std}}\) over all such pairings. Let \(W\) be the \(d\)-by-\(w\) block matrix reversing each pairing, with \(W_{\mu\lambda}\) the corresponding sum of \(E_{lk}^{\mathrm{std}}\). It sends all unpaired coordinates with \(k>d_0\) to zero. Directly on these coordinates, \[WV=I_d\otimes I_s,\qquad VW=I_w\otimes\Pi.\] Define block matrices \[J^{(\theta)}=(I_w\otimes Q)V,\qquad Y^{(\theta)}=W(I_w\otimes Q^{-1}).\] They have outer shapes \(w\times d\) and \(d\times w\), respectively, with entries in \(\operatorname{Mat}_s(D)\). Their products are \[Y^{(\theta)}J^{(\theta)}=I_d\otimes I_s,\qquad J^{(\theta)}Y^{(\theta)}=I_w\otimes E.\] Finally define the \((\lambda,\mu)\) entry of \(J_i\) by applying \(\rho^{-1}\) to the tuple \((J^{(\theta)}_{\lambda\mu})_\theta\), and similarly define \(Y_i\) entrywise from the \(Y^{(\theta)}\). The isomorphism (19) carries \(f\) to the tuple of identity matrices and \(P_i\) to the tuple of matrices \(E\). The last two identities therefore pull back to (16). This proves Proposition 9.

All choices in the matrix construction are over finite sets: character orbits, coset representatives, augmentation matrices, and coordinate pairings. Taking the first element or the first independent columns in fixed orders makes the prescription definite. The inverse (21) is a finite sum, and the inverse (18) is a finite group-algebra formula. Thus the matrices furnished by the proposition are finite sums in the explicitly attached finite groups.

We now apply Proposition 9 to the embedded groups of Proposition 8 and the identical idempotents of Proposition 4. The group \(G_0\) is countable because it is finitely generated. All the ranks equal the positive integer \(d\) from Equation (7). Hence the group \(G_1\), the nonzero central idempotent \(f\), and the matrices \(J_i,Y_i\) are now fixed by the constructions above. Corollary 6 also gives \(P_iP_j=0\) on every edge with \(i<j\). These are precisely the inputs to the next section.

Finite prescriptions and scope.

Every choice of character, basis, field element, or finite inverse occurs in an explicitly finite set. Products of the resulting group words define finite group-algebra sums even when different words happen to coincide. No word comparisons are needed to define the elements, and the kernel certificate and injective matrix block certify the nonzero reverse defect. The countable permutation actions used to certify group embeddings are not coefficient-search inputs.

Containing torsion does not say that every element has finite order. The group must be infinite, since finite-dimensional left multiplication proves direct finiteness for every finite group algebra, in modular characteristic as well as otherwise. The coefficient field throughout remains the specified \(K\) of order \(p^4\).

More generally, the supports of any two finite sums in a group algebra generate a finitely generated subgroup. Its group algebra embeds by the group bases, preserving both a one-sided product and its nonzero reverse defect. Thus a universal direct-finiteness theorem for finitely generated groups would already imply the assertion for all groups.