Tilings of symmetric and alternating groups by conjugacy classes

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≥5, 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≥5, 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≥ 5) not containing the identity element cannot tile.

Categorical and Floer-theoretic entropies: comparison and counting conjugacy classes in symplectic mapping class groups
Sangjin Lee
arXiv3 Nov 2026PreprintarXiv licenceFull textDynamical systems Geometry & topologyarXiv
Conjugacy and period in symmetric diagram groups
Matteo Tarocchi
arXiv4 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Finite groups with conjugacy classes of size 2pⁿ
Antonio Beltran, Changguo shao
arXiv13 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
On the number of conjugacy classes of finite solvable groups of order divisible by p²
Thomas Michael Keller
arXiv9 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Large orbits of p-groups on conjugacy classes
Thomas Michael Keller
arXiv4 Nov 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{somlai2026tilings,
  title = {{Tilings of symmetric and alternating groups by conjugacy classes}},
  author = {Gábor Somlai and Binzhou Xia and Sanming Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01267},
  eprint = {2610.01267},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01267},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Tilings of symmetric and alternating groups by conjugacy classes
AU  - Gábor Somlai
AU  - Binzhou Xia
AU  - Sanming Zhou
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01267
UR  - https://arxiv.org/abs/2610.01267
AB  - 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≥5, 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≥5, 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≥ 5) not containing the identity element cannot tile.
ER  -

Abstract

Related

Cite