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REVIEW 3 major objections 5 minor 25 references

Generalised shuffle groups

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Generalized perfect shuffles are proved to generate the full symmetric or alternating group in three infinite families.

desk verdict Solid group theory paper that confirms Medvedoff-Morrison for three infinite families and introduces generalized shuffle groups; the main weakness is the repeated reliance on undocumented Magma checks for load-bearing finite cases. read the letter →

arxiv 1908.05128 v1 pith:XGT4VOOO submitted 2019-08-14 math.GR

Carmen Amarra , Luke Morgan , Cheryl E. Praeger This is my paper · ORCID
classification math.GR MSC 20B2505E18
keywords cardshufflingperfectshufflespermutationgroupsprimitiveshuffleproductactionaffinecascading
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

This paper studies a many-handed generalization of perfect shuffles: a deck of $kn$ cards is cut into $k$ piles of $n$ and all $k!$ pile permutations are allowed, giving a group $\operatorname{Sh}(\operatorname{Sym}(k), n)$ inside the symmetric group $\operatorname{Sym}(kn)$. A long-standing conjecture says that this shuffle group is as large as possible — the full symmetric group or the alternating group — whenever $n$ is not a power of $k$ (with a small exclusion at $k=4$). The paper proves that conjecture for three infinite families: all $k>n$, all pairs $(k,n)=(\ell^e,\ell^f)$ with $e\nmid f$ and $k\neq 4$, and all $k=2^e\ge 4$ with $n$ not a power of $2$. It also sets up a broader theory of shuffle groups $\operatorname{Sh}(P,n)$ for arbitrary pile-permutation groups $P$, proving primitivity and 2-transitivity results that may apply well beyond the conjecture.

What carries the argument

The load-bearing construction is the shuffle $\sigma$ itself: on the $kn-1$ nonzero card positions it acts as multiplication by $k$ modulo $kn-1$, fixing the top and bottom cards (Lemma 2.2). In the power case $k=\ell^e$, $n=\ell^f$, writing cards in base $\ell$ identifies the deck with the set $[\ell]^{e+f}$ of $(e+f)$-tuples, under which $\sigma$ becomes the cyclic shift of coordinates by $e$ places and the pile-permutation group becomes a wreath product in product action — the group that permutes the coordinates of each tuple independently and then permutes the coordinates themselves. This yields $\operatorname{Sh}(P,\ell^f) = P \wr C_{1+f/e}$ when $e\mid f$, and the full symmetric or affine group when $e\nmid f$ (Theorem 1.4), the affine group being the transformations $x\mapsto xA+b$ of a vector space over the prime field. For $k>n$, the key tools are 2-transitivity of $\operatorname{Sh}(P,n)$ for 2-transitive $P$, minimal-degree bounds (Bochert's bound and the minimal degree of a wreath product in product action), the dichotomy that a finite 2-transitive group is either affine or almost simple, and Zsigmondy's theorem on primitive prime divisors to eliminate classical candidates. For $k=2^e$, the cascade groups $G_t=\operatorname{Sh}(V_t,2^{e-t}n)$ with $V_t$ the elementary abelian group of order $2^t$ — the translations of a $t$-dimensional vector space over the two-element field — form a nested chain; the base $G_1$ is known from earlier work, and the equalities among the $G_t$ force the overgroup $\operatorname{Sh}(\operatorname{Sym}(2^e), n)$ to contain the alternating group.

What would settle it

Independently recompute $\operatorname{Sh}(\operatorname{Sym}(k), n)$ for each pair $2\le n<k\le 14$ (and, say, for $(k,n)=(5,3)$) in a second computer algebra system and check that the order is $|\operatorname{Alt}(kn)|$ or $|\operatorname{Sym}(kn)|$ except for $(4,2)$, where the order should be that of $\operatorname{AGL}(3,2)$; a single mismatch would falsify Theorem 1.8(1).

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Extended reading notes

Core claim

The central discovery is that the generalized shuffle group $\operatorname{Sh}(\operatorname{Sym}(k), n)$ equals $\operatorname{Alt}(kn)$ or $\operatorname{Sym}(kn)$ in three doubly infinite families: whenever $k>n\ge 2$ (with the single exception $(k,n)=(4,2)$, where it is $\operatorname{AGL}(3,2)$, the affine group on a 3-dimensional vector space over the field of 2 elements); whenever $k=\ell^e$ and $n=\ell^f$ with $f$ not a multiple of $e$ and $k\neq 4$; and whenever $k=2^e\ge 4$ and $n$ is not a power of $2$. The alternatives are decided by parity: the group lies in $\operatorname{Alt}(kn)$ exactly when $n\equiv 0 \pmod 4$, or $n\equiv 2 \pmod 4$ and $k\equiv 0$ or $1 \pmod 4$, or $n$ is odd and the pile group lies in $\operatorname{Alt}(k)$; otherwise it is $\operatorname{Sym}(kn)$. The proof passes through a structural theorem for 'power case' shuffle groups (where $k$ and $n$ are powers of the same integer), a general primitivity theorem for primitive pile groups, a full analysis of the $k>n$ 2-transitive case using the classification of finite 2-transitive groups, and a cascade argument showing that shuffle groups with elementary abelian pile groups often coincide.

Load-bearing premise

Everything rests on finite group computations performed with the Magma computer algebra system and reported without the underlying scripts or output logs: the verification for all pairs $2\le n<k\le 14$, the elimination of the fifteen almost simple candidates in Table 3, and the three exceptional cascade cases $(4,3)$, $(4,6)$ and $(8,3)$ — if any of those computations is wrong, the corresponding theorem is not established.

Editorial extensions

If this is right

  • For $k>n$, the shuffle group is the full symmetric or alternating group, so the dealer can achieve every permutation of the deck (or every even permutation).
  • The structure of $\operatorname{Sh}(\operatorname{Sym}(2^e), n)$ is now completely known for every $n$: combine Theorem 1.2 for $n=2^f$, the affine result for $k=4$ and $n=2^f$, and Corollary 1.10 otherwise.
  • The conjecture is confirmed for all prime-power pile counts $k=\ell^e$ when the deck count $n$ is also a power of $\ell$ with exponent not dividing $e$; in the prime case this yields the affine group $\operatorname{AGL}(e+f,\ell)$.
  • Any counterexample to the conjecture must have $n\ge k$, so future work can focus on the regime where piles are at least as large as their number.
  • If $P$ is a 2-transitive almost simple group (a group between a nonabelian finite simple group and its automorphism group) and $k>n$, then $\operatorname{Sh}(P,n)$ is almost simple; if $P$ is affine and $n$ is not a prime power of the same prime, $\operatorname{Sh}(P,n)$ contains $\operatorname{Alt}(kn)$.

Reading between the lines

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

  • The cascade mechanism suggests an analogue for odd primes: if the structure of $\operatorname{Sh}(\operatorname{Sym}(3), m)$ were known for all $m$, the same nesting argument could confirm the conjecture for $k=3^e$ and $n$ not a power of $3$ — the paper explicitly notes this is the missing ingredient.
  • The computational evidence that even the cyclic pile group $C_k$ generates $\operatorname{Alt}(kn)$ for $k\le 13$ and $n\le 1000$ hints that the 'as large as possible' phenomenon may hold far beyond $\operatorname{Sym}(k)$; if so, the full symmetric group is not needed to mix the deck completely.
  • The theorem that $\operatorname{Sh}(P,n)$ is primitive for primitive non-regular $P$ gives a route to classify all pairs $(P,M)$ where $\operatorname{Sh}(P,n)$ is trapped in a maximal subgroup $M$ — the question the paper poses as Question 1.11 — because the only maximal overgroups to worry about are the primitive ones.
  • If the conjecture is true for all remaining pairs, then the minimal degree of $\operatorname{Sh}(\operatorname{Sym}(k),n)$ should be $2n$ (from the transposition of two piles), a quantity one could test computationally for small $k,n$ to detect a counterexample before constructing the whole group.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the generalised shuffle group Sh(P,n) ≤ Sym(kn) generated by the standard k-pile out-shuffle and the pile permutations from P, following Medvedoff-Morrison and Diaconis-Graham-Kantor. The main result is confirmation of Medvedoff-Morrison's Conjecture 1.3 for three doubly infinite families: (i) k>n (Theorem 1.8(1)), (ii) k=ℓ^e, n=ℓ^f with e∤f and k≠4 (Corollary 1.5), and (iii) k=2^e with n not a power of 2 (Corollary 1.10). Along the way the paper proves structure theorems for Sh(P,n) in the 'power case' (Theorem 1.4), primitivity (Theorem 1.6), and 2-transitivity results (Theorem 1.8).

Significance. The results are substantial, confirming a long-standing conjecture in broad families and providing a general framework for shuffle groups with arbitrary pile permutation groups. The analytic development is coherent: the power-case identification (Section 3), the primitivity proof using orbital digraphs (Theorem 1.6), and the classification analysis via Burnside and the classification of 2-transitive groups (Section 5) are all carefully argued. The paper also introduces the cascading shuffle group technique (Section 6), which is elegant. However, the completeness and verifiability of the main theorems rests on several asserted finite computations in Magma, for which no scripts or logs are supplied.

major comments (3)
  1. [Section 5, Lemma 5.2 and Theorem 5.10] The proof of Theorem 1.8(1) for k>n is completed by Lemma 5.2, where the remaining cases 2≤n<k≤14 are checked 'by computer' via Magma, and Theorem 5.10 eliminates the fifteen candidates in Table 3 using Magma. These finite checks are load-bearing: they are the only support for the exclusion of the almost simple candidates in Theorem 5.10 and for the k≤14 cases of Lemma 5.2. Without the scripts, input files, and output logs, a reader cannot verify that the computations used the generators of Definition 2.1, that no cases were missed, or that the reported equalities are correct. Please provide the actual Magma code and logs (or an independent implementation in GAP) for these checks, including the exact commands and the verification of the group equalities and containments.
  2. [Sections 5 and 6 (Lemma 5.3, Lemma 5.4, Theorem 1.9, Corollary 1.10)] The proofs also invoke Magma for specific exceptional cases: PSU(4,2) and PSL(2,7), PSL(2,8) in Lemma 5.3; the sporadic candidates in Lemma 5.4; and the cases (4,3), (4,6), (8,3) in Theorem 1.9 and Corollary 1.10. These cases are essential for the cascading family k=2^e and for the affine/almost-simple dichotomy. Please document these computations as well, or replace them by short human-verifiable arguments; as it stands, a mistake in any one of these finite checks could invalidate the corresponding headline family.
  3. [Section 3, Proposition 3.3(6)] The statement reads 'if f ∤ e and T = Sym(e), then G ∩ Y = Sym(e + f)', but the proof and the application in Theorem 1.4 require the condition 'e ∤ f' (i.e., f is not a multiple of e). As written, the statement is false: for e = 2, f = 4, T = Sym(2), Proposition 3.3(5) gives G ∩ Y ≅ Sym(2) ≀ C_3, not Sym(6). Please correct the condition; the proof in the text already uses the correct hypothesis.
minor comments (5)
  1. [Section 2.1, Lemma 2.7] The proof of the minimal degree of a wreath product is correct, but the formula in the semiregular case could be stated more explicitly as d^c - d^{c-1}; the current notation is a little terse.
  2. [Section 6, Lemma 6.1(b)] The notation (ve−t+s)ρ is ambiguous; it should be written as (v_{e-t+s})^ρ to make the subscript clear.
  3. [Throughout] The manuscript contains numerous OCR artifacts such as '/greaterorequalslant', '/n⋊tless⋊rslnteql' and similar; these should be cleaned up in the final version.
  4. [Section 1.3] The sentence 'we have shown that the conjecture holds, that is, that Sh(Sym(k), n) ≥ Alt(kn)' is imprecise; the conjecture also specifies the exact parity cases, so the wording should be tightened.
  5. [References] Reference [5] is cited as an online PDF; if a published or stable version exists, it should be cited.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected: the confirmation of Conjecture 1.3 rests on external classification theorems and explicit generator computations, with Magma used only for finite base cases.

full rationale

The paper never defines or derives Sh(Sym(k),n) from Conjecture 1.3; rather it proves that Sh(Sym(k),n) contains Alt(kn) using independent tools: Diaconis-Graham-Kantor Theorem 1.1, Medvedoff-Morrison Theorem 1.2, Burnside's 2-transitive dichotomy, the CFSG-based classification of 2-transitive groups (Cameron's Table 7.4), Zsigmondy's theorem, Guralnick's prime-power index theorem, and the Liebeck-Praeger-Saxl maximal subgroup classification. The product identification in Section 3 and the primitivity argument in Section 4 are self-contained derivations from the definition of Sh(P,n). The cascading argument in Section 6 derives the subgroup relations among G1,...,Ge from explicit generator identities in Lemma 6.1, then uses an external maximal subgroup theorem (Wielandt) to pass from Ge to Sh(Sym(k),n). The finite Magma checks (Lemma 5.2, Lemma 5.4, Theorem 5.10, and the exceptional cases in Theorem 1.9/Corollary 1.10) are finite verifications of concrete group equalities and containments, not fitted parameters, renamed predictions, or assumed versions of the target result. The absence of scripts or logs is a reproducibility gap, not circularity. Self-citations to [16], [18], and [20] are to established published theorems used as external evidence, and they do not assume Conjecture 1.3 or the paper's own conclusions. No equation in the paper reduces to its own input, and no load-bearing claim is justified solely by an unverified self-citation.

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

The central claim rests on the standard classification apparatus for finite permutation groups, on classical bounds (Bochert, Guralnick, Zsigmondy), and on unshipped Magma verifications of finitely many exceptional cases. No free parameters or invented entities occur.

assumptions (5)
  • standard math CFSG-based classification of 2-transitive groups and maximal subgroups of Alt/Sym (Cameron; Liebeck-Praeger-Saxl)
    Invoked in Lemma 5.2, Theorem 5.10, and Corollary 1.5 to enumerate possible 2-transitive socles and to show certain subgroups are maximal.
  • standard math Bochert's minimal degree bound for 2-transitive groups not containing Alt(m)
    Used in Lemma 5.2 to reduce the k>n family to finitely many cases k≤14.
  • standard math Zsigmondy's theorem and Lemma 2.9 p-part calculations
    Used in Section 5 (Lemmas 5.6-5.9) to constrain prime-power parameters in the almost simple cases.
  • domain assumption Magma verifications for finite exceptional cases are correct
    The paper states 'By Magma' and 'we used Magma' without code or logs; these checks support Lemma 5.2, Theorem 5.10, and the special cases in Theorem 1.9.
  • standard math Known structure of Sh(Sym(2), m) (Diaconis-Graham-Kantor) and Sh(Sym(4), 2^f) (Cohen et al.)
    Used as the base case G1 in the cascading argument in Section 6.

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Pith. "Pith review of Generalised shuffle groups." pith.science (2026). https://pith.science/paper/XGT4VOOO

@misc{pith2026190805128,
  author       = {Pith},
  title        = {Pith review of: Generalised shuffle groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XGT4VOOO}},
  note         = {Machine review of arXiv:1908.05128}
}
abstract

The mathematics of shuffling a deck of $2n$ cards with two "perfect shuffles" was brought into clarity by Diaconis, Graham and Kantor. Here we consider a generalisation of this problem, with a so-called "many handed dealer" shuffling $kn$ cards by cutting into $k$ piles with $n$ cards in each pile and using $k!$ shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, so long as $k\neq 4$ and $n$ is not a power of $k$. We confirm this conjecture for three doubly infinite families of integers: all $(k,n)$ with $k>n$; all $(k, n)\in \{ (\ell^e, \ell^f )\mid \ell \geqslant 2, \ell^e>4, f \ \mbox{not a multiple of}\ e\}$; and all $(k,n)$ with $k=2^e\geqslant 4$ and $n$ not a power of $2$. We open up a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles.

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Bochert, Ueber die Classe der transitiven Substitutionen-gr uppen, Math

    A. Bochert, Ueber die Classe der transitiven Substitutionen-gr uppen, Math. Annalen 40 (1892), 176– 193

    work page

  2. [2]

    Bray, Derek F

    John N. Bray, Derek F. Holt, Colva M. Roney-Dougal. The maximal subgroups of the low-dimensional finite classical groups. With a foreword by Martin Liebeck. London M athematical Society Lecture Note Series, 407. Cambridge University Press, Cambridge, (2013)

    work page 2013

  3. [3]

    Wieb Bosma, John Cannon, and Catherine Playoust, The Magma alg ebra system. I. The user lan- guage, J. Symbolic Comput. , 24 (1997), 235–265

    work page 1997

  4. [4]

    Cameron, Permutation Groups

    Peter J. Cameron, Permutation Groups. London Mathematical Society Student Texts, 45. Cambridge University Press, Cambridge, (1999)

    work page 1999

  5. [5]

    Morrison, Sarah Wright , Perfect shuffles and affine groups

    Amanda Cohen, Andr´ e Harmse, Kent E. Morrison, Sarah Wright , Perfect shuffles and affine groups . online: https://www.calpoly.edu/ ∼kmorriso/Research/shuffles.pdf

    work page

  6. [6]

    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson. An ATLAS of Finite Groups. Clarendon Press, Oxford, 1985; reprinted with correct ions 2003

    work page 1985

  7. [7]

    J. -A. de S´ eguier, Groupes de substitutions , Gauthier-Villars, 1912, Paris

    work page 1912

  8. [8]

    Persi Diaconis, R. L. Graham, William M. Kantor, The mathematics of perfect shuffles . Adv. in Appl. Math. 4 (1983), no. 2, 175–196

    work page 1983

Show all 25 references
  1. [9]

    Dixon, The Probability of Generating the Symmetric Group , Math

    John D. Dixon, The Probability of Generating the Symmetric Group , Math. Z. 110 (1969), 199–205

    1969

  2. [10]

    J. D. Dixon and B. Mortimer, Permutation Groups. Springer-Verlag, New York, 1996

    1996

  3. [11]

    Estes, R

    D. Estes, R. Guralnick, M. Schacher, E. Straus. Equations in prime powers . Pacific J. Math. 118 (1985), no. 2, 359–367

    1985

  4. [12]

    Huppert, Endliche Gruppen

    B. Huppert, Endliche Gruppen . I. Die Grundlehren der Mathematischen Wissenschaften, Band 134 Springer-Verlag, Berlin-New York (1967)

    1967

  5. [13]

    Guralnick, Subgroups of Prime Power Index in a Simple Group

    Robert M. Guralnick, Subgroups of Prime Power Index in a Simple Group . J. Alg. 81 (1983), 304–311

    1983

  6. [14]

    Kleidman, The Maximal Subgroups of the Chevalley Groups G2(q) with q Odd, the Ree Groups 2G2(q), and Their Automorphism Groups , J

    Peter B. Kleidman, The Maximal Subgroups of the Chevalley Groups G2(q) with q Odd, the Ree Groups 2G2(q), and Their Automorphism Groups , J. Algebra 117 30–71 (1988)

    1988

  7. [15]

    The Subgroup Structure of the Finite Classical Groups

    Peter Kleidman and Martin Liebeck. The Subgroup Structure of the Finite Classical Groups . (London Mathematical Society Lecture Note Series). Cambridge: Cambridg e University Press. (1990)

    1990

  8. [16]

    Martin Liebeck, Cheryl Praeger, Jan Saxl, A Classification of the Maximal Subgroups of the Finite Alternating and Symmetric Groups . J. Alg. 111 (1987) 365–383

    1987

  9. [17]

    Martin Liebeck, Jan Saxl, Minimal degrees of primitive permutation groups, with an ap plication to monodromy groups of covers of Riemann surfaces . Proc. London Math. soc. (3) 63 (1991) 266–314

    1991

  10. [18]

    L¨ ubeck, A

    F. L¨ ubeck, A. C. Niemeyer, and C. E. Praeger, Finding involutions in finite Lie type groups of odd characteristic, J. Algebra 321 (2009), 3397–3417

    2009

  11. [19]

    Steve Medvedoff, Kent Morrison, Groups of perfect shuffles . Math. Mag. 60 (1987), no. 1, 3–14

    1987

  12. [20]

    A. C. Niemeyer, T. Popiel, and C. E. Praeger, Abundant p-singular elements in finite classical groups , J. Algebra 408 (2014), 189–204

    2014

  13. [21]

    Praeger, The inclusion problem for finite primitive permutation grou ps

    Cheryl E. Praeger, The inclusion problem for finite primitive permutation grou ps. Proc. London Math. Soc. 60 (1990), 68–88

    1990

  14. [22]

    American J

    Michio Suzuki, On Finite Groups with Cyclic Sylow Subgroups for All Odd Prim es. American J. Math. 77 no. 4 (1955), 657–691

    1955

  15. [23]

    Michio Suzuki, A new type of simple groups of finite order . Proc. Nat. Acad. Sci. U.S.A. 46 (1960) 868–870

    1960

  16. [24]

    Academic Press (1964)

    Helmut Wielandt, Finite Permutation Groups . Academic Press (1964)

    1964

  17. [25]

    Zsigmondy, Zur Theorie der Potenzreste , Monatsh

    K. Zsigmondy, Zur Theorie der Potenzreste , Monatsh. F¨ ur Math. u. Phys. 3 (1892) 265–284. Carmen Amarra, Institute of Mathematics, University of the Philippines Diliman, C. P. Garcia A venue, Diliman, Quezon City 1101, Philippines E-mail address : mcamarra@math.upd.edu.ph 33...

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