Schur multipliers of special p-groups of rank 2

Sumana Hatui

REVIEW 4 major objections 4 minor 12 references

Schur multipliers of special p-groups of rank 2

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper determines the exact order and structure of the Schur multiplier for every special p-group of rank 2, completing the classification problem for these groups.

desk verdict The rank-2 Schur multiplier computation is new and probably right, but Theorem 1.4's capable-group classification leans on a suspicious citation to a p^6 table for p^5 groups and on an unreleased preprint. read the letter →

arxiv 1908.06409 v1 pith:NGR32VPI submitted 2019-08-18 math.GR

Sumana Hatui This is my paper · ORCID

classification math.GR MSC 20J9920D15

keywords Schurmultiplierfinitep-groupspecialp-groupsrank2capablegroupsepicenterextraspecialclass

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

Special p-groups of rank 2 are finite p-groups whose commutator subgroup and centre coincide, both being an elementary abelian group of order $p^{2}$, with elementary abelian quotient by the commutator. The paper shows that for such a group G with d generators, the order of the Schur multiplier M(G) is completely controlled by d and by the subgroup G^p generated by p-th powers. Depending on whether G^p equals the commutator subgroup, is cyclic of order p, or is trivial, |M(G)| is one of a short explicit list of powers of p, ranging from $p^{{d(d-1)/2-2}}$ to $p^{{d(d-1)/2+3}}$. In all cases except one the multiplier is elementary abelian; in the exceptional case its exponent is at most $p^{2}$. The proof also identifies exactly which groups in the family are capable, i.e. arise as H/Z(H), and this answers the open problem listed as Problem 2027 in the standard reference on groups of prime power order.

What carries the argument

The argument rests on an explicit model of M(G) for class-2 groups: write V=G/G' and W=G'=Z(G) as vector spaces over F_p, let (v_1,v_2)=[g_1,g_2] be the commutator bilinear map, and let f(v)=g^p be the p-power map. The multiplier M(G) is isomorphic to a subgroup M of an abelian extension M^*, and its order satisfies |M(G)| = $p^{{d(d-1)/2-2}}$ \cdot $p^{{2d}}$/|X|, where X is the subspace of V \otimes W generated by the triple-commutator elements v_1 \otimes (v_2,v_3)+v_2 \otimes (v_3,v_1)+v_3 \otimes (v_1,v_2) and the p-power elements v \otimes f(v). The paper's main technical work is to compute the dimension of X in each of the three cases G^p=G', G^p \cong Z_p, and G^p=1, using a basis of generators; the exact sequence for M(G) and M(G/Z) plus the criterion identifying the epicenter Z^*(G) then convert those dimensions into multiplier orders and capability statements.

What would settle it

Compute M(G) directly for the $p^{7}$ group T defined in Theorem 1.4(f), with five x-generators x_1,\dots,x_5 and relations [x_2,x_1]=[x_5,x_3]=c_1 and [x_3,x_1]=[x_5,x_4]=c_2; the paper predicts |M(T)|=$p^{9}$ for every odd p. If an independent computation from the presentation yields any other p-power, the central claim fails. A less expensive check is to verify that the five groups in Theorem 1.4(c) really exhaust all capable groups of order $p^{6}$ by comparing the indicated multiplier orders with a direct enumeration from the published list of groups of order $p^{6}$.

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Extended reading notes

Core claim

Let G be a special p-group of rank 2 with d=d(G). The central result is a complete computation of M(G). If G^p=G' \cong Z_p \times Z_p, then M(G) is elementary abelian of order $p^{{d(d-1)/2-2}}$ exactly when Z^*(G)=Z(G), and otherwise G is capable and |M(G)|=$p^{{d(d-1)/2-1}}$ with exponent at most $p^{2}$; in this case every quotient G/Z by a central subgroup of order p is a product of an extraspecial p-group with an elementary abelian group. If G^p \cong Z_p, then G is never capable, M(G) is elementary abelian, and the only possible orders are $p^{{d(d-1)/2-2}}$ and $p^{{d(d-1)/2}}$, each characterized by whether the epicenter is Z(G) or G^p and by the isomorphism type of G/G^p. If p is odd and G^p=1, then M(G) is elementary abelian, its order lies between $p^{{d(d-1)/2-2}}$ and $p^{{d(d-1)/2+3}}$, and the capable groups are exactly five isomorphism types with explicit presentations: one of order $p^{5}$, three of order $p^{6}$, and one of order $p^{7}$. The extreme multiplier orders are attained exactly on these types, with the smallest order characterizing Z^*(G)=Z(G). For p=2 the paper proves that $G^{2}$=G' necessarily, so the first case covers every special 2-group of rank 2.

Load-bearing premise

The load-bearing premise is that the cited classifications of capable groups of orders $p^{5}$, $p^{6}$, and $p^{7}$ are complete, so the enumeration of the five capable isomorphism types in Theorem 1.4(c) misses no group; if one of those classifications is incomplete, the matching multiplier orders would be incomplete as well.

Editorial extensions

If this is right

For any special p-group of rank 2, the Schur multiplier — both its order and, in nearly all cases, its elementary abelian structure — is determined by just two data: the number of generators and the subgroup of p-th powers; no other invariant is needed.
The only non-elementary multiplier that can occur has exponent at most p^2 and order p^{d(d-1)/2-1}, and it occurs precisely for capable groups with G^p=G'.
If G^p is trivial and p is odd, a group in this family is capable if and only if it is one of the five listed isomorphism types; capability is therefore a restricted, order-small phenomenon inside the family.
For p=2, no special 2-group of rank 2 has G^2 of order p or trivial; the case G^2=G' already covers the whole family, so the answer for p=2 is contained in the first theorem.
The formulas give sharp bounds: the exponent of |M(G)| always lies in the interval [d(d-1)/2-2, d(d-1)/2+3].

Reading between the lines

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

The dimension-counting method is not obviously tied to rank 2, so one can test whether a similar formula governs special p-groups of rank k; the main difficulty would be the growth of the triple-commutator span X_1 as k increases.
The theorem supplies a clean dataset for testing general conjectures connecting |M(G)| with d(G) and |G'| for finite p-groups, since here both parameters vary and the answer is exact.
Because capability is characterized so concretely in this family — five groups when G^p=1 and a dichotomy in the other cases — one could use these groups as building blocks to probe which central extensions H realize them as H/Z(H).
For p=2, the nonexistence of special 2-groups of rank 2 with G^2 \cong Z_2 or G^2=1 may hint at a broader phenomenon linking the p-power map to the commutator structure in class-2 groups; this is an editorial extrapolation, not a claim of the paper.

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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

This paper answers Berkovich–Janko Problem 2027 by computing the Schur multiplier of special p-groups of rank 2. The main results are new: earlier work covered extraspecial groups (rank 1) and maximum-rank special groups; this fills the rank-2 case. The approach is sensible—use the Blackburn–Evens construction, the Ganea sequence, and existing classifications, then split on G^p (order p^2, p, or 1). Theorems 1.1 and 1.3, which handle G^p=G' and G^p≅Z_p, are mostly transparent and the order formulas look plausible. The list of capable groups in Theorem 1.4(c) is the payoff.

The soft spots are concentrated in the proof of Theorem 1.4. For |G|=p^5, the proof says 'looking through the list of groups given in [7]'—but [7] is James's table of groups of order p^6. A p^5 group is not in that list, so the uniqueness of Φ4(15) does not follow from the citation as written. Maybe Heineken's Proposition 3 plus a central-quotient argument supplies the missing step, but it isn't given. That is a genuine audit gap, not a demonstrated error. The order-p^7 case is also hard to check because it depends on the unpublished preprint [6] with no reproduced verification. The p=2 argument in Theorem 1.5 is terse; the contradiction using Theorems 2.6–2.8 could be spelled out, but it's not obviously wrong.

So the stress-test note lands: the most load-bearing classification step in Theorem 1.4 is not checkable from the text as written. The rest of the paper, especially Theorems 1.1 and 1.3, is in better shape.

This is a real contribution to finite p-group theory, but the exposition of Theorem 1.4 needs serious repair. I would recommend peer review, with a request for major revision: supply the missing p^5 argument, give more detail for order p^7 (or reproduce the needed part of [6]), and expand the p=2 reasoning. The referee should be someone who knows capable p-groups and the James/Heineken classifications.

Referee Report

4 major / 4 minor

Summary. The paper studies the Schur multiplier M(G) of special p-groups of rank 2, i.e., groups with G' = Z(G) elementary abelian of order p^2 and G/G' elementary abelian. The main results give explicit formulas for |M(G)| in terms of d = d(G), according to the order of the subgroup G^p: when G^p = G' (Theorem 1.1), when G^p is cyclic of order p (Theorem 1.3), and when G^p = 1 with p odd (Theorem 1.4). Theorem 1.4 also contains a classification of capable groups in the exponent-p case, listing five isomorphism types, and Theorem 1.5 asserts that every special 2-group of rank 2 satisfies G^2 = G'. The proofs use the Blackburn--Evens construction of M(G) as a subgroup of a group M* determined by the commutator map, the Ganea map, and external classifications of p-groups with prescribed Schur multiplier orders.

Significance. If the results are correct, they give a complete answer to Berkovich--Janko Problem 2027 for special p-groups of rank 2: explicit powers of p for the order of the Schur multiplier, a sharp range in the exponent-p case, and a concrete list of capable groups. The dichotomy between Z*(G) = Z(G) and Z*(G) = 1 is clean, and the bound in Theorem 1.4(b) is concrete and testable. The paper also demonstrates a reasonable method: reducing |M(G)| to the size of the obstruction space X via equation (2.2) and then bounding X by direct linear-algebra arguments. The main limitation is that the capable-group classification in Theorem 1.4(c) is not self-contained, and the proof of Theorem 1.5 for p = 2 appears to contain a genuine gap.

major comments (4)

[Section 3, proof of Theorem 1.4, order p^5 case] The proof states 'If |G|=p^5, then looking through the list of groups given in [7], it follows that G ≅ Φ4(15).' However, reference [7] is R. James's table of groups of order p^6, not p^5. A group of order p^5 cannot be identified by inspecting a list of groups of order p^6 unless an additional argument is supplied. Since the exceptional multiplier order in part (d), p^{d(d-1)/2 + 3}, depends on this uniqueness claim, the classification of capable groups of order p^5 is not established by the cited source.
[Section 3, proof of Theorem 1.4, order p^7 case] The unique capable group T of order p^7 is taken from the unpublished preprint [6] with no statement or verification of the classification result. Because Theorem 1.4(c) is an 'if and only if' assertion, the dependence on an unreviewed preprint makes the exhaustiveness check impossible from the text. Please include the precise classification result from [6], restate its proof, or replace it with a self-contained argument.
[Section 3, proof of Theorem 1.4, after the capable-group list] The size computations for X are asserted without derivation. Specifically, 'Since |X|=p' for Φ4(15) and 'Since |X|=p^4' for Φ12(16), Φ13(16), Φ15(16), and the implied value for T, are not shown. These values are load-bearing: via (2.2) they determine the multiplier orders in parts (d)-(f). Please provide the actual computation of X, or at least a basis and dimension count, for each of the five capable groups.
[Section 3, proof of Theorem 1.5] The proof of Theorem 1.5 is not sound as written. The step 'By Theorem 1.3, it follows that there is no special 2-group of rank 2 with G^2 ≅ Z_2' is not justified, because Theorem 1.3 as stated does not exclude p = 2; the proof of Theorem 1.3(b) simply asserts 'By (c) and (d) it follows that p must be odd,' which is unexplained. In the subsequent contradiction argument for G^2 = 1, the inequality k-m ≤ -1 is not a contradiction; for example k=0 and m=1 yields k-m = -1. Therefore the conclusion that no special 2-group with G^2 = 1 exists does not follow. This leaves the p = 2 case of the main problem unproven.

minor comments (4)

[Title] The manuscript title contains a typo: 'MUL TIPLIERS' should read 'MULTIPLIERS'.
[Section 2, note after Theorem 2.1] In the displayed generator set for X1, the expression '[x1.x2]' should be '[x1, x2]'.
[Throughout] The notation for extraspecial groups is inconsistent: Theorem 1.1(d) uses ESp_2(p^{2m+1}) and ESp_2(p^3), while Theorem 1.4 uses ESp(p^3) and ESp(p^{2m+1}) without subscripts. Please define the notation once and use it uniformly.
[Section 3, proof of Theorem 1.4] In the proof of part (b), the equality |X| = |X1| is used but not justified; it follows from G^p = 1 only after noting that f = 0 and hence X2 = 0. This should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation rests on external classification theorems and the Blackburn–Evens machine, not on assumptions equivalent to the conclusions.

full rationale

The paper computes Schur multipliers of special p-groups of rank 2 using the Blackburn–Evens construction, Ganea's exact sequence, and previously established multiplier classifications (Niroomand, Blackburn–Evens, etc.). The central formula (2.2) derives |M(G)| from |X|, where X is built from the group's own commutator and p-power structure, and |X| is bounded directly in each case. The capable-group lists in Theorem 1.4 are imported from external sources: Heineken's Proposition 3 [5], the Heineken–Kappe–Morse preprint [6], and James's table [7]. These are independent citations, not self-citations, and none of them is equivalent to the paper's multiplier computation. No fitted parameter is relabeled as a prediction, and no definition presupposes the target result. The only notable weakness is a correctness/verification issue in the proof of Theorem 1.4: for |G|=p^5, the text says 'looking through the list of groups given in [7]' although [7] classifies groups of order p^6, so the uniqueness of Φ4(15) is not fully transparent from the cited source. This is a support gap, not a circular step, and per the hard rules it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests on a chain of external classifications and the Blackburn-Evens machine. No new objects or fitted parameters are introduced; the central risk is the correctness and accessibility of the cited classifications, especially the preprint [6].

assumptions (5)

standard math Blackburn-Evens construction identifies M(G) with the group M built from V, W, X1, X2 and sigma (Theorem 2.1 = [3, Theorem 3.1]).
Invoked in Section 2 and used throughout; gives the formula |M(G)| = p^(d(d-1)/2 - 2) * p^(2d)/|X|.
standard math Ganea exact sequence (Theorem 2.2) and the Beyl-Felgner-Schmid characterization of the epicenter (Theorem 2.3).
Used to relate |M(G)| to |M(G/Z)| and to characterize Z*(G) via the subspace X.
standard math Niroomand classification theorems for p-groups with certain Schur multiplier orders (Theorems 2.6, 2.7, 2.8).
Used to identify quotient groups G/Z from Schur multiplier sizes.
domain assumption Heineken's Proposition 3 and the Heineken-Kappe-Morse classification of capable special p-groups of rank 2 ([5], [6]).
Used in Theorem 1.4(c)-(e) to enumerate capable groups of orders p^5, p^6 and p^7; [6] is cited as a preprint.
domain assumption James's list of groups of order p^6 for odd p ([7]).
Used by inspection to identify Phi4(15), Phi12(16), Phi13(16) and Phi15(16).

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Cite this review

Pith. "Pith review of Schur multipliers of special p-groups of rank 2." pith.science (2026). https://pith.science/paper/NGR32VPI

@misc{pith2026190806409,
  author       = {Pith},
  title        = {Pith review of: Schur multipliers of special p-groups of rank 2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGR32VPI}},
  note         = {Machine review of arXiv:1908.06409}
}

read the original abstract

A group G is called special p-group of rank k if the commutator subgroup [G,G] and centre Z(G) are equal, which is elementary abelian p-group of rank k and G/[G,G] is also elementary abelian p-group. In this article we determine the Schur multiplier of special p-groups of rank 2 explicitly.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[7]

James, The groups of order p6 (p an odd prime) , Math

R. James, The groups of order p6 (p an odd prime) , Math. Comp. 34 (1980), 613-637

work page 1980
[6]

Heineken, L.C

H. Heineken, L.C. Kappe, R.F. Morse, On the classiﬁcation of special p-groups of rank two that appear as central quotient groups , Preprint

work page
[1]

Berkovich, Z

Y. Berkovich, Z. Janko, Groups of Prime Power Order , vol. 3, W alter de Gruyter, 2011

work page 2011
[2]

F. R. Beyl, U. Felgner, P. Schmid, On groups occurring as center factor groups , J. Algebra 61 (1979), 161-177

work page 1979
[3]

Blackburn, L

N. Blackburn, L. Evens, Schur multipliers of p-groups , J. Reine Angew. Math. 309 (1979), 100-113

work page 1979
[4]

Ganea, Homologie et extensions centrales de groupes , C

T. Ganea, Homologie et extensions centrales de groupes , C. R. Acad. Sci., Paris 266 (1968), 556-568

work page 1968
[5]

Heineken, Nilpotent groups of class two that can appear as central quot ient groups , Rend

H. Heineken, Nilpotent groups of class two that can appear as central quot ient groups , Rend. Sem. Mat. Univ. Padova 84 (1990), 241-248

work page 1990
[8]

Karpilovsky, The Schur multiplier

G. Karpilovsky, The Schur multiplier . London Math. Soc. Monographs, Oxford Univ. Press, 1987

work page 1987

Show all 12 references

[9]

Niroomand, On the order of Schur multiplier of non-abelian p groups , Journal of Algebra, 322 (2009), 4479-4482

P. Niroomand, On the order of Schur multiplier of non-abelian p groups , Journal of Algebra, 322 (2009), 4479-4482

2009
[10]

Niroomand, A note on the Schur multiplier of groups of prime power order , Ricerche di Matematica, 61 (2012), 341-346

P. Niroomand, A note on the Schur multiplier of groups of prime power order , Ricerche di Matematica, 61 (2012), 341-346

2012
[11]

Niroomand, Classifying p-groups by their Schur multipliers , Math

P. Niroomand, Classifying p-groups by their Schur multipliers , Math. Rep. (Bucur.) 20(70) (2018), no. 3, 279-284

2018
[12]

P. K. Rai, On the Schur multiplier of special p-groups, J. Pure Appl. Algebra, 222 (2018), 316-322. Department of Mathematics, Indian Institute of Science (II Sc), Bangalore-560012, India E-mail address : sumanahatui@iisc.ac.in, sumana.iitg@gmail.com

2018

Pith tools

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

[1] [7]

James, The groups of order p6 (p an odd prime) , Math

R. James, The groups of order p6 (p an odd prime) , Math. Comp. 34 (1980), 613-637

work page 1980

[2] [6]

Heineken, L.C

H. Heineken, L.C. Kappe, R.F. Morse, On the classiﬁcation of special p-groups of rank two that appear as central quotient groups , Preprint

work page

[3] [1]

Berkovich, Z

Y. Berkovich, Z. Janko, Groups of Prime Power Order , vol. 3, W alter de Gruyter, 2011

work page 2011

[4] [2]

F. R. Beyl, U. Felgner, P. Schmid, On groups occurring as center factor groups , J. Algebra 61 (1979), 161-177

work page 1979

[5] [3]

Blackburn, L

N. Blackburn, L. Evens, Schur multipliers of p-groups , J. Reine Angew. Math. 309 (1979), 100-113

work page 1979

[6] [4]

Ganea, Homologie et extensions centrales de groupes , C

T. Ganea, Homologie et extensions centrales de groupes , C. R. Acad. Sci., Paris 266 (1968), 556-568

work page 1968

[7] [5]

Heineken, Nilpotent groups of class two that can appear as central quot ient groups , Rend

H. Heineken, Nilpotent groups of class two that can appear as central quot ient groups , Rend. Sem. Mat. Univ. Padova 84 (1990), 241-248

work page 1990

[8] [8]

Karpilovsky, The Schur multiplier

G. Karpilovsky, The Schur multiplier . London Math. Soc. Monographs, Oxford Univ. Press, 1987

work page 1987