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arxiv: 2010.04954 · v1 · submitted 2020-10-10 · 🧮 math.GR · math.CO

Powers in the wreath product of G with S_n

Rijubrata Kundu , Sudipa Mondal This is my paper

Pith reviewed 2026-05-24 14:28 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords wreath productr-th powerspower mapprobabilityconjugacy classessymmetric groupfinite groups
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The pith

If the order of G is coprime to prime r, the probability that a random element of G wr S_n is an r-th power equals the probability for G wr S_{n+1} except when n ≡ -1 mod r.

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

The paper examines the r-th power map on the wreath product of any finite group G with the symmetric group S_n. It defines P_r as the fraction of elements that arise as r-th powers. When |G| shares no common factors with r, this fraction stays constant from one n to the next provided n is not congruent to -1 modulo r. The authors also supply an explicit count of the conjugacy classes consisting entirely of r-th powers. These results describe how powers distribute inside an important class of permutation groups built from G.

Core claim

For prime r ≥ 2 and finite G with |G| coprime to r, the equality P_r(G wr S_{n+1}) = P_r(G wr S_n) holds for every n ≢ -1 (mod r). A formula is also given for the number of conjugacy classes of r-th powers inside G wr S_n.

What carries the argument

The r-th power map ω_r on the wreath product G wr S_n, whose image size yields the probability P_r.

If this is right

  • The relative size of the set of r-th powers is independent of n in the stated range.
  • The number of conjugacy classes consisting of r-th powers admits an explicit formula in terms of the conjugacy classes of G and the partitions of n.
  • The result applies to every finite group G whose order is coprime to the given prime r.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stabilization may hold for other natural actions or for the full monomial group over G.
  • The class-count formula could be combined with the class equation to obtain the proportion of r-powers without enumerating elements.
  • One could check whether the coprimeness hypothesis can be dropped when G is abelian or when r is fixed and G varies.

Load-bearing premise

The order of G shares no prime factors with r.

What would settle it

Direct computation of the size of the image of the r-power map in both G wr S_1 and G wr S_2 for any G and prime r where 1 ≢ -1 mod r and |G| coprime to r; the two probabilities must match if the claim holds.

read the original abstract

In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\geq 2$, a prime, consider $\omega_r: G\wr S_n\to G\wr S_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|\omega_r(G\wr S_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\not \equiv -1(\text{mod } r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript defines P_r(G wr S_n) as the proportion of r-th powers in the wreath product G wr S_n (r prime) and proves that this probability satisfies P_r(G wr S_{n+1}) = P_r(G wr S_n) whenever n ≢ -1 (mod r) and gcd(|G|, r) = 1. It also supplies an explicit formula for the number of conjugacy classes in G wr S_n that consist entirely of r-th powers.

Significance. If the central claims hold, the stabilization result gives a clean description of the image size of the r-power map once n is large enough relative to r, under the coprimeness hypothesis that makes the base-group map bijective. The conjugacy-class formula supplies a concrete enumerative tool for wreath-product class functions. Both results are of interest in the study of word maps on imprimitive groups.

minor comments (1)
  1. [Abstract] Abstract: the phrasing 'We prove, P_r(G wr S_{n+1})=P_r(G wr S_n)' contains an extraneous comma after 'prove'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no points to address point-by-point at this stage. We remain available to incorporate any minor changes the referee or editor may identify.

Circularity Check

0 steps flagged

No circularity: theorem proved from group action properties under explicit coprimeness hypothesis

full rationale

The manuscript states and proves an equality P_r(G wr S_{n+1}) = P_r(G wr S_n) for n ≢ -1 mod r when gcd(|G|,r)=1, together with a formula for r-power conjugacy classes. No equations reduce the claimed probability or class count to a fitted parameter or self-referential definition; the coprimeness hypothesis is an external assumption used to ensure bijectivity on the base group, not a derived quantity. No self-citations appear in the abstract or described derivation chain, and the result is not obtained by renaming a known empirical pattern or smuggling an ansatz. The derivation is therefore self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure mathematics proof relying on standard properties of finite groups and wreath products; no fitted parameters or invented entities are indicated.

axioms (1)
  • standard math Standard axioms and basic properties of finite groups, wreath products, and conjugacy classes
    The statements about powers and probabilities presuppose the usual definitions and theorems of group theory.

pith-pipeline@v0.9.0 · 5700 in / 1132 out tokens · 19833 ms · 2026-05-24T14:28:05.584527+00:00 · methodology

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Reference graph

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16 extracted references · 16 canonical work pages

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