{"id":"14eecd6d-9940-4650-8bd1-75569e26e11f","arxiv_id":"1908.03877","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For finite abelian p-groups, for 2-groups of maximal class, and for 2-groups of order at most 16, the classical unitary subgroup of the modular group algebra determines the group.","lead":"This paper proves that for certain finite p-groups, the subgroup of group-algebra units fixed by the classical involution determines the original group. The result extends the modular isomorphism problem by showing that a much smaller subgroup of units still remembers the group.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's proof never shows V_*(FG) determines |G|; it only separates the three maximal-class isomorphism types at a fixed order, so the 'only if' direction is incomplete across different 2-powers.","rationale":"The reader's weakest_assumption identifies exactly the same gap: Theorem 2's proof separates the three maximal-class groups at a fixed order but never proves that V_*(FG) determines |G|. I considered other possible concerns, including Theorem 3's reliance on the unreleased RAMEGA package and an old technical report, but the cross-order gap in Theorem 2 is more load-bearing because Theorem 2 is a headline result and its only-if direction is incomplete as written. The paper itself signals the issue by saying, after Lemma 4, that a similar statement 'seems to be true' for non-abelian algebras rather than proving it. Since this is a proof gap rather than a demonstrated counterexample, the conditional verdict is appropriate; no change to the reader's verdict is needed.","tokens_in":11266,"tokens_out":5872,"duration_ms":64747,"concrete_test":"Derive an explicit expression for |V_*(FG)|, or at least for the involution count Theta_G(2), for G = D_{2n+1}, Q_{2n+1}, D^-_{2n+1} as a function of n, and prove that the three families are pairwise disjoint and each is injective in n; this is the missing cross-order step of Theorem 2. As a first finite check, compute these invariants for n = 2,3,4,5 using the generating sets in [12,13] or the RAMEGA package, and test whether any two maximal-class groups of different orders yield the same involution count; if even one cross-order collision appears, the involution-count argument cannot prove the theorem without an additional invariant.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing problem is in the proof of Theorem 2. The theorem quantifies over all finite maximal-class 2-groups G,H over F_2. The proof, however, fixes n in presentations (2) and shows inequality (9): Theta_Q(2) < Theta_SD(2) < Theta_D(2) for the number of involutions in V_*(FG) (Lemmas 8-9 plus the D^- argument). This is an invariant that separates the three groups only at one fixed order 2^{n+1}. To conclude V_*(FG) isomorphic to V_*(FH) implies G isomorphic to H for arbitrary G,H, one must also rule out |G| different from |H|. No such argument is given: there is no formula for |V_*(FG)| as a function of n for these groups, no comparison of Theta across different n, and no other invariant recovering n. Lemma 4, which says |V_*(FG)| determines |G|, is proved only for abelian G; the text immediately adds that a 'similar statement seems to be true' for non-abelian group algebras, which is not a proof. Consequently the only-if direction of Theorem 2 is not established for pairs of different orders, and the theorem as stated is not fully proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the *-unitary isomorphism problem (*-UIP): for a finite p-group G over a finite field F of characteristic p, does the normalized unitary subgroup V_*(FG) of the modular group algebra determine G up to isomorphism? The authors give affirmative answers for finite abelian p-groups (Theorem 1), for 2-groups of maximal class over the field of 2 elements (Theorem 2), and for non-abelian 2-groups of order at most 16 over F_2 (Theorem 3). The proofs combine known structural descriptions of V_*(FG) with involution-counting arguments, and Theorem 3 is supported by computational verification with the GAP package RAMEGA.","tokens_in":11522,"tokens_out":12623,"duration_ms":110998,"significance":"The *-UIP is a natural strengthening of the classical modular isomorphism problem, and positive results for these classes would be a meaningful contribution to the study of unitary subgroups of modular group algebras. The paper also provides detailed counts of involutions in V_*(FG) for dihedral, quaternion, and semidihedral groups, which are of independent interest. The computational verification of Theorem 3 is a strength. However, the proof of Theorem 2 is incomplete as written, so the central claim for maximal-class 2-groups is not fully established; this tempers the significance of the paper until the gap is addressed.","major_comments":[{"comment":"The proof of Theorem 2 does not establish the 'only if' direction for maximal-class groups of different orders. Inequality (9), Theta_Q(2) < Theta_{D^-}(2) < Theta_D(2), is derived for a fixed n in the presentations (2), so it separates the three isomorphism types only within one order 2^{n+1}. To conclude that V_*(FG) is isomorphic to V_*(FH) implies G is isomorphic to H for arbitrary maximal-class G and H, the proof must also show that the isomorphism class of V_*(FG) determines |G|; no such invariant is computed in Section 4. Lemma 4 supplies this only for abelian G, and the text immediately after Lemma 4 explicitly says that the non-abelian analogue only 'seems to be true'. Thus the theorem as stated is not fully proved. The authors should either prove that |V_*(FG)| or another invariant determines the order of G for the dihedral, quaternion, and semidihedral groups, or restrict Theorem 2 to groups of a fixed order.","section":"Section 4, Proof of Theorem 2, inequality (9)"}],"minor_comments":[{"comment":"The displayed formula in Lemma 1(iii) has a minor typo: '|V*(F G|' is missing a closing parenthesis; it should read '|V_*(FG)|' or similar.","section":"Section 2, Lemma 1(iii)"},{"comment":"The final simplification in Lemma 8 appears to be incorrect as displayed: for n = 4 the expression 2^{2n+2} - 2^{3*2^{n-2}+1} is negative, which cannot be a count of involutions. Please reconcile the final line with the preceding computation, which includes the additional term 2^{2n+1}.","section":"Section 4, Lemma 8"},{"comment":"The exponent notation in equation (3) for |V_*(FC)| is ambiguous; please write the exponent explicitly, for example as 2^{2^{n-1}+2}, so that the order is unambiguous.","section":"Section 4, equation (3)"},{"comment":"The statements of Theorems 1 and 2 use the phrase 'for some group H'; since V_*(FH) is only defined for finite p-groups, the theorems should specify that H is a finite p-group (or that V_*(FH) is defined).","section":"Section 1, Theorems 1 and 2"},{"comment":"The notation 'D8 Y C4' in the list S is not defined in the paper; please define the operation (for instance, whether it denotes a central product or a semidirect product).","section":"Section 4, Proof of Theorem 3"},{"comment":"The proof does not explicitly compare the orders of V_*(FG) for the order-8 groups (which are 2^6) with those for the order-16 groups (which are at least 2^10); adding a sentence making this cross-order comparison would make the 'only if' direction fully explicit.","section":"Section 4, Proof of Theorem 3"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the gap in Theorem 2: the proof separates the three maximal-class types at a fixed order but does not recover the order itself. If the authors can supply the missing order-recovery argument, for example by showing that |V_*(FG)| is a strictly increasing function of n for these groups, the paper would be acceptable. The reliance on previous structural results, including several from the same research group, is not circular but would benefit from independent verification, especially the generator sets cited from [12]. The GAP verification of Theorem 3 is a positive signal. I recommend asking for a major revision rather than rejection, because the gap is local and likely fixable within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zsolt,\n\nHere's my read. The paper answers the *-unitary isomorphism problem affirmatively for two classes where the answer wasn't previously known: finite abelian p-groups (Theorem 1) and the three maximal-class 2-group families (Theorem 2, partially), plus a computational check for non-abelian 2-groups of order at most 16 (Theorem 3). The novelty is real, though the machinery is mostly imported from earlier structural work on unitary subgroups; the isomorphism statements themselves are new.\n\nWhat it does well: Lemma 8 and Lemma 9 are genuine computations. The involution-counting for the dihedral and quaternion cases is coherent once you parse the notation, and Theorem 1 is a clean repackaging of known structural results for abelian groups. The paper is honest about its reliance on RAMEGA and the old Bovdi–Erdei technical report.\n\nThe soft spots, in order of severity:\n\n1. Theorem 2, as the stress-test correctly notes, never shows that V_*(FG) determines |G|. The proof fixes n and compares Θ_Q < Θ_SD < Θ_D at that fixed order. It does not rule out isomorphisms between V_*(FG) for different 2-powers. Lemma 4 is proven only for abelian groups, and the 'similar statement seems to be true' is not a proof. This is a load-bearing gap in the 'only if' direction. A revision needs a cross-order invariant or a weakened statement.\n\n2. Theorem 3 depends on an unreleased GAP package and a 1996 technical report. The generator lists are detailed, but several pairwise non-isomorphism claims are not fully verified in the paper. Reproducibility is a concern, not a fatal one.\n\n3. Minor: Lemma 4's order-determination argument skips the explicit observation that the exponent intervals [|G|/2, |G|-1] are disjoint across different |G|. Also, the introduction says 'necessary to prove only the if part' — the arrow is backwards, a harmless typo.\n\nWho it's for: specialists in modular group algebras. The abelian theorem is citable; Theorem 2 in current form should be cited with caution. The paper deserves serious refereeing. An editor should send it to an expert, expecting that Theorem 2 either gets repaired or restated.\n\nMy recommendation: accept for peer review, but the referee should push on the cross-order issue before publication.","headline":"Solid but incomplete: Theorem 2's central 'only if' direction does not cross group orders, while Theorems 1 and 3 are serviceable; the paper warrants expert peer review with expectations of repair.","tokens_in":11998,"tokens_out":4255,"would_cite":false,"duration_ms":44165,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16U60","20D15","20C05","16S34"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the normalized unitary subgroup $V_*(FG)$ of the modular group algebra determines the finite $p$-group $G$ up to isomorphism for three large classes: abelian $p$-groups, maximal-class 2-groups over $\\mathbb{F}_2$…","keywords":["group algebra","unitary subgroup","isomorphism problem","modular group algebra","finite p-groups","2-groups of maximal class","classical involution","unit group"],"falsifier":"Compute, over $\\mathbb{F}_2$, the order and the number of involutions of $V_*(FG)$ for the dihedral, semidihedral, and generalized quaternion groups of two distinct orders; a single cross-order equality in either invariant would break the only-if proof of Theorem 2, and an explicit isomorphism $V_*(FG) \\cong V_*(FH)$ for non-isomorphic maximal-class groups $G,H$ would refute the theorem as stated.","tokens_in":11076,"feed_emoji":"🔢","tokens_out":14362,"duration_ms":139530,"temperature":0.7,"pith_summary":"The paper asks a sharper version of the classical isomorphism problem for modular group algebras: instead of asking whether the full normalized unit group determines the group, it asks whether the much smaller normalized unitary subgroup $V_*(FG)$ — the units $u$ with $u^{-1}=u^*$, where $*$ is the involution sending each group element to its inverse — already determines $G$. It establishes that $V_*(FG) \\cong V_*(FH)$ implies $G \\cong H$ for three families: finite abelian $p$-groups over any finite field of characteristic $p$, finite 2-groups of maximal class over the field of two elements, and nonabelian 2-groups of order at most 16 over the same field. The converse direction is immediate, since an isomorphism of the underlying groups induces an isomorphism of their unitary subgroups. If the claims are right, a comparatively small portion of the unit group of $FG$ encodes the isomorphism type of $G$.","feed_headline":"The unitary subgroup of a group algebra determines the p-group","feed_subtitle":"Proofs cover abelian p-groups, maximal-class 2-groups, and 2-groups up to order 16.","key_machinery":"The object that carries the whole argument is $V_*(FG)$, the normalized unitary subgroup of the modular group algebra $FG$ with respect to the classical involution $g \\mapsto g^{-1}$. In the abelian case the decisive tool is the structure formula [14, Theorem 2] expressing $V_*(FG)$ as a direct product of cyclic $p$-groups, with the numbers of cyclic factors of each height written as linear combinations of $|G^{p^i}|$, $|G^{p^i}[2]|$, and $f_i(G)$; inverting these formulas recovers $G$. In the maximal-class case the decisive tool is the involution count $\\Theta_G(2) = |\\{x \\in V_*(FG) : x^2 = 1\\}|$, obtained by solving the equations $x^2=1$ and $x^*=x$ inside the group algebra of a cyclic subgroup $C$, and the strict inequalities $\\Theta_Q < \\Theta_{D^-} < \\Theta_D$ do the separation. In the order-16 case the machinery is the catalogue of explicit generators for $V_*(FG)$ for each nonabelian group of that order, taken from [12] and [13], compared by elementary invariants.","core_discovery":"The central claim is that $V_*(FG)$ is a complete isomorphism invariant for the stated classes. In the abelian case the proof works by showing that the invariant-factor decomposition of $V_*(FG)$, supplied by a structure theorem from [14], determines the numbers $f_i(G)$ of cyclic factors of each order in $G$; once those numbers are known, the isomorphism type of the finite abelian $p$-group follows. In the maximal-class case the proof counts involutions in $V_*(FG)$ for the three possible types — dihedral, generalized quaternion, and semidihedral — and obtains the strict ordering $\\Theta_Q < \\Theta_{D^-} < \\Theta_D$ within each fixed order, so the involution count tells the three types apart. For nonabelian groups of order at most 16 over $\\mathbb{F}_2$, the proof goes case by case through the finite list of groups, comparing explicit presentations of their unitary subgroups by order, commutator subgroup, and whether the unitary subgroup is Hamiltonian (nonabelian, with all subgroups normal).","pith_inferences":["The same order-plus-involution strategy suggests a testable path for larger 2-groups: list the groups of a given order, compute $|V_*(FG)|$ and the involution count for each, and check injectivity of those invariants; wherever injective, the conclusion follows without a full presentation of $V_*(FG)$.","The abelian proof reconstructs $G$ from the $f_i(G)$ via a linear system, so it could be turned into an explicit algorithm: compute the invariant-factor decomposition of $V_*(FG)$ and invert the formulas from [14].","The proof of Theorem 2 leaves it implicit that $V_*(FG)$ determines the order of a maximal-class 2-group; if that order-determination step is supplied, the involution-count inequalities would settle all maximal-class orders at once, and if it fails, a counterexample may be found across different orders.","The Hamiltonicity criterion used at order 16 hints that $V_*(FG)$ may also register other structural properties of $G$, such as having a cyclic derived subgroup; testing this at order 32 is a natural next step."],"forward_implications":["For finite abelian $p$-groups, the isomorphism type of $G$ is recoverable from the direct-product decomposition of the unitary subgroup $V_*(FG)$ alone.","For 2-groups of maximal class over $\\mathbb{F}_2$, a single numerical invariant — the number of involutions in $V_*(FG)$ — separates dihedral, semidihedral, and generalized quaternion groups of the same order.","For nonabelian 2-groups of order at most 16 over $\\mathbb{F}_2$, the unitary subgroup distinguishes every group in the class, so the *-unitary isomorphism problem has an affirmative answer in this range.","In all three classes, deciding whether two groups are isomorphic can in principle be done by inspecting $V_*(FG)$ rather than the full unit group $V(FG)$."],"supporting_citations":[{"why":"Supplies the structure formula for $V_*(FG)$ as a direct product of cyclic 2-groups, with rank formulas in terms of subgroup counts; this is the engine of Theorem 1.","marker":"[14]"},{"why":"Gives the sets $H_i$ inside $V(FC)$ and their sizes, which the involution-counting argument for maximal-class 2-groups uses.","marker":"[5]"},{"why":"Provides explicit generator sets for $V_*(FG)$ for nonabelian groups of order 16, used case by case in Theorem 3.","marker":"[12]"},{"why":"Supplies the structure of unitary subgroups of modular group algebras of 2-groups, including order-16 and maximal-class cases; used in Theorems 2 and 3.","marker":"[13]"},{"why":"Gives the criterion that $V_*(FG)$ is Hamiltonian exactly when $G$ is Hamiltonian, used in the order-16 classification.","marker":"[6]"},{"why":"Characterizes when the unitary subgroup equals the full normalized unit group for abelian 2-groups, used to dispose of small cases in Theorem 1.","marker":"[10]"},{"why":"Provides the annihilator bound that yields $|A_i|=2^i$ in the involution-count computation of Lemma 8.","marker":"[24]"}],"fun_headline_variants":["Unitary subgroup determines the p-group","Isomorphism problem solved: unitary subgroup is the key","Unitary subgroup singles out the p-group","Group algebra's unitary subgroup fixes the group"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that $V_*(FG)$ determines the order of a maximal-class 2-group; the proof of Theorem 2 only compares groups of one fixed order and never shows that groups of different orders cannot have isomorphic unitary subgroups.","fun_headline_variants_meta":{"raw":{"variants":["Unitary subgroup determines the p-group","Isomorphism problem solved: unitary subgroup is the key","Unitary subgroup singles out the p-group","Group algebra's unitary subgroup fixes the group"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000752,"raw_usage":{"total_tokens":3289,"prompt_tokens":833,"completion_tokens":2456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":2400}},"tokens_in":449,"tokens_out":2456,"duration_ms":18728,"temperature":1.0,"reasoning_tokens":2400,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:02:01.402788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, over $\\mathbb{F}_2$, the order and the number of involutions of $V_*(FG)$ for the dihedral, semidihedral, and generalized quaternion groups of two distinct orders; a single cross-order equality in either invariant would break the only-if proof of Theorem 2, and an explicit isomorphism $V_*(FG) \\cong V_*(FH)$ for non-isomorphic maximal-class groups $G,H$ would refute the theorem as stated.","supporting_citations":[{"cited_title":"Bovdi and A","cited_arxiv_id":null,"evidence_quote":"Supplies the structure formula for $V_*(FG)$ as a direct product of cyclic 2-groups, with rank formulas in terms of subgroup counts; this is the engine of Theorem 1."},{"cited_title":"Balogh and A","cited_arxiv_id":null,"evidence_quote":"Gives the sets $H_i$ inside $V(FC)$ and their sizes, which the involution-counting argument for maximal-class 2-groups uses."},{"cited_title":"Bovdi and L","cited_arxiv_id":null,"evidence_quote":"Provides explicit generator sets for $V_*(FG)$ for nonabelian groups of order 16, used case by case in Theorem 3."},{"cited_title":"Bovdi and L","cited_arxiv_id":null,"evidence_quote":"Supplies the structure of unitary subgroups of modular group algebras of 2-groups, including order-16 and maximal-class cases; used in Theorems 2 and 3."},{"cited_title":"Balogh, L","cited_arxiv_id":null,"evidence_quote":"Gives the criterion that $V_*(FG)$ is Hamiltonian exactly when $G$ is Hamiltonian, used in the order-16 classification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterizes when the unitary subgroup equals the full normalized unit group for abelian 2-groups, used to dispose of small cases in Theorem 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the annihilator bound that yields $|A_i|=2^i$ in the involution-count computation of Lemma 8."}],"review_version":1}