Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.01267

Tilings of symmetric and alternating groups by conjugacy classes

Gábor Somlai·Binzhou Xia·Sanming Zhou
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study tilings of symmetric and alternating groups by conjugacy classes. For $S_n$, the case where the conjugacy class consists of transpositions was first investigated by Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. Our proofs combine representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \iffalse We study the problem of whether a conjugacy class tiles the symmetric group or alternating group. For $S_n$, the case where the conjugacy class consists of transpositions was first investigated by Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. The proof combines representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \fi \iffalse In this paper, we study when a conjugacy class or a normal subset tiles the symmetric or alternating group. The case of transpositions was first investigated by Rothaus and Thompson, and in general, it remains open. We show that every other conjugacy class of a symmetric group does not tile. We also prove that normal subsets in $A_n$ ($n\ge 5)$ not containing the identity element cannot tile.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gábor Somlai, Binzhou Xia, Sanming Zhou. 2026-10-01. Tilings of symmetric and alternating groups by conjugacy classes. https://arxiv.org/abs/2610.01267

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

JSJ splittings for all Artin groups

We prove that an Artin group splits over infinite cyclic subgroups if and only if its defining graph has a separating vertex, and explicitly construct a JSJ decomposition over infinite cyclic subgroups for all Artin groups. We then use these facts to show that, if two Artin groups are isomorphic, then they have the same set of parabolics supported on "big chunks", that is, maximal subgraphs without separating vertices. We also deduce acylindrical hyperbolicity for the automorphism groups of many Artin groups, partially answering a question of Genevois in the case of Artin groups. As a consequence, we produce new families of Artin groups with the R-infinity property.

math.GR↗
arXiv2026

Finite-valued invariant metrics and a classification of natural groups

For every group $G$, of arbitrary cardinality, we construct a right-invariant metric with at most $32$ values whose isometries are exactly the permutations preserving every right-invariant metric on $G$. The proof combines subgroup-entry ranks and sign-variation colorings with a short-word rigidity theorem of Leemann and de la Salle. Their nonabelian orientation-rigidity theorem and direct regular-subgroup arguments yield the complete classification of natural groups in the right-translation sense: an abelian group $A$ is natural if and only if $2A=A$ or $2A=\{0\}$, and a nonabelian group is natural if and only if it is not generalized dicyclic. In particular, the additive group of every field is natural. The bound improves to $17$ for abelian groups and $5$ for Boolean groups, and the Boolean bound is sharp: $C_2^3$ admits no such metric with fewer than five values. Complementary constructions give one countable-valued hull metric realizing precisely the affine sign isometries simultaneously on all subgroups containing fixed coordinate markers, and signed-basis metrics with at most $p+5$ values over $\mathbb{F}_p$ for odd $p$. No other bound is claimed optimal, and no uncolored graphical regular representation is asserted.

math.GR↗
arXiv2026

The Margolis-Rhodes Monoid of a Graph

We investigate the structural, combinatorial, ideal-theoretic and Krohn-Rhodes complexity of the Margolis-Rhodes monoid, MR(G), of a finite simple graph G, viewed topologically as a 1-dimensional simplicial complex. Alongside the full monoid, we examine some subsemigroups including St(G), defined by the condition that the full inverse image is an edge or the empty set and Inj(G), the monoid of all partial 1-1 continuous functions. We provide explicit combinatorial enumerations and struture for paths and cycles. We compute Green's relations showing in particular that the partial order of regular J-classes is isomorphic to the poset of induced subgraphs of G. Finally, we apply these structural invariants to Krohn-Rhodes complexity theory. It is known that the Margolis-Rhodes monoid has complexity at most 2 and 1 for St(G) and Inj(G). We show that for cycles the complexity of its Margolis-Rhodes monoid is 2 if and only if the cycle is of length at least 4. For paths, we prove that the complexity of its Margolis-Rhodes monoid is 2 if the path length is at least 13.

math.GR↗