Isomorphism problem of Unitary Subgroups of Group Algebras

Victor Bovdi; Zsolt Balogh

REVIEW 1 major objections 6 minor 32 references

Isomorphism problem of Unitary Subgroups of Group Algebras

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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 $\mathbb{F}_2$…

desk verdict 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. read the letter →

arxiv 1908.03877 v2 pith:COSNZSRW submitted 2019-08-11 math.RA math.GRmath.RT

Zsolt Balogh , Victor Bovdi This is my paper · ORCID

classification math.RAmath.GRmath.RT MSC 16U6020D1520C0516S34

keywords groupalgebraunitarysubgroupisomorphismproblemmodularfinitep-groups2-groupsofmaximalclassclassicalinvolutionunit

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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 $G$.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

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 $V(FG)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Zsolt,

Here'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.

What 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.

The soft spots, in order of severity:

1. 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.

2. 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.

3. 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.

Who 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.

My recommendation: accept for peer review, but the referee should push on the cross-order issue before publication.

Referee Report

1 major / 6 minor

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.

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 (1)

[Section 4, Proof of Theorem 2, inequality (9)] 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.

minor comments (6)

[Section 2, Lemma 1(iii)] 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 4, Lemma 8] 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 4, equation (3)] 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 1, Theorems 1 and 2] 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 4, Proof of Theorem 3] 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 4, Proof of Theorem 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorems are derived from independent structural descriptions and explicit involution counts, not from the conclusions they purport to prove.

full rationale

The derivation chain is not circular. Theorem 1 is obtained from Lemma 1 ([14]) and Lemma 2's arithmetical reconstruction of the Hall invariants |G^{p^i}| of an abelian p-group from the invariants f_i(V_*(FG)); the theorem's target conclusion 'V_*(FG) ≅ V_*(FH) iff G ≅ H' is not used as an input to any of these lemmas. Theorem 2 computes the number of involutions Θ_G(2) = |V_*(FG)[2]| for D_{2^{n+1}}, Q_{2^{n+1}}, and D^-_{2^{n+1}} using systems (5), (8), (10) and Lemma 6 from [5], an explicit count of the sets H_i in the cyclic subalgebra F C. Lemma 6 is parameter-free, has stated assumptions that do not contain the target result, and has a published proof; citing it is independent support rather than circular import. Theorem 3 is a case-by-case comparison using published generator presentations of V_*(FG) for order-16 groups ([12], [13], [18], [19], [23]) and a GAP/RAMEGA verification; those presentations do not assume the *-isomorphism problem. No equation is used as both input and output, and no fitted parameter is renamed as a prediction. The only flagged weakness is not circular: after Lemma 4 the authors write 'A similar statement seems to be true for non-abelian group algebras.' In Theorem 2, the proof fixes n and establishes inequality (9) among the three maximal-class groups of order 2^{n+1}, but it never proves that V_*(FG) determines |G| for non-abelian maximal-class groups, so the 'only if' direction for groups of different orders is not fully established. This is a completeness/correctness gap, not a self-referential reduction of the claim to its own input, and therefore it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on no fitted constants and no newly invented algebraic objects. The main external inputs are published structural descriptions of unitary subgroups, several from the same research group but not circular with the target theorem.

assumptions (4)

domain assumption Finite field of characteristic p and finite p-group G are the setting; the classical involution sends g to g^{-1}.
This is the problem setup of the paper and is not a new postulate.
domain assumption Known structural results for V_*(FG): Lemma 1 from [14], Lemma 6 from [5], generator sets for order-16 groups from [12,13], and unit-group structures from [18,19].
Theorems 1 to 3 rely on these external descriptions; they are not proved inside the paper.
standard math For a p-group over a field of characteristic p, the group algebra is local and every nonunit in F C has the form gamma (1+a)^i with gamma a unit.
Used in Lemmas 8 and 9 when classifying nonunit components x2 of involutions.
domain assumption Maximal-class 2-groups are exactly the dihedral, semidihedral, and generalized quaternion groups.
Invoked at the start of Section 4 as a standard classification theorem.

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Cite this review

Pith. "Pith review of Isomorphism problem of Unitary Subgroups of Group Algebras." pith.science (2026). https://pith.science/paper/COSNZSRW

@misc{pith2026190803877,
  author       = {Pith},
  title        = {Pith review of: Isomorphism problem of Unitary Subgroups of Group Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/COSNZSRW}},
  note         = {Machine review of arXiv:1908.03877}
}

read the original abstract

Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bagi´ nski

C. Bagi´ nski. The isomorphism question for modular grou p algebras of metacyclic p-groups. Proc. Amer. Math. Soc. , 104(1):39–42, 1988

work page 1988

[2] [2]

Bagi´ nski

C. Bagi´ nski. Modular group algebras of 2-groups of maxi mal class. Comm. Algebra, 20(5):1229–1241, 1992

work page 1992

[3] [3]

Bagi´ nski and A

C. Bagi´ nski and A. Caranti. The modular group algebras o f p-groups of maximal class. Canad. J. Math. , 40(6):1422–1435, 1988

work page 1988

[4] [4]

Balogh and A

Z. Balogh and A. Bovdi. Group algebras with unit group of c lass p. Publ. Math. Debrecen , 65(3-4):261–268, 2004

work page 2004

[5] [5]

Balogh and A

Z. Balogh and A. Bovdi. On units of group algebras of 2-gro ups of maximal class. Comm. Algebra, 32(8):3227–3245, 2004

work page 2004

[6] [6]

Balogh, L

Z. Balogh, L. Creedon, and J. Gildea. Involutions and uni tary subgroups in group algebras. Acta Sci. Math. (Szeged) , 79(3-4):391– 400, 2013

work page 2013

[7] [7]

Balogh and V

Z. Balogh and V. Laver. Isomorphism problem of unitary su bgroups of group algebras. Accepted in Ukrainian Math. J

work page

[8] [8]

Balogh and V

Z. Balogh and V. Laver. RAMEGA – RAndom MEthods in Group Al gebras, Version 1.0.0. 2020

work page 2020