Subindices of two-element subsets of finite groups

Abstract

Let A be a nonempty subset of a group G. A right subfactor relative to A is a subset B maximal under inclusion among those for which every element of AB has a unique representation as ab, with a∈ A and b∈ B. Left subfactors are defined analogously using BA. The infima and suprema of the cardinalities of right and left subfactors define four \emph{subindices}. A subset is called \emph{index-stable} if its four subindices coincide. For any infinite group G, every nonempty finite subset of G is index-stable. For finite groups, the situation is more complicated. We prove that for a two-element subset of a finite group, the left and right subindices coincide pairwise. Let A={x,y} with x≠ y, and let X be the cyclic subgroup generated by x⁻¹y. Then the lower subindex of A is |G:X|·⌈ |X|/3⌉, and the upper subindex is |G:X|·⌊ |X|/2⌋. These formulas imply that A is index-stable if and only if |X| is one of $2,3,4,5,7$. For any two-element subset of a finite group G, the lower subindex is at least ⌈ |G|/3⌉, and the upper subindex is at most ⌊ |G|/2⌋. We establish necessary and sufficient conditions for equality in each of these bounds. All two-element subsets of a finite group are index-stable if and only if every element of the group has order $1$, $2$, $3$, $4$, $5$, or $7$. We give a complete structural classification of such groups.

Subindices and subfactors of ℤ_n and k-index stability of finite groups
M. H. Hooshmand
arXiv18 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Finite groups with large power-avoiding subsets
Simon R. Blackburn, Sarah B. Hart, Daniel McVeagh
arXiv16 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Element Orders and Character Codegrees of Finite Groups
Yong Yang
arXiv6 Sep 2026PreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv
Finite groups with large sum of element orders compared to cyclic groups
Péter P. Pálfy, Alex D. Simonyi
arXiv20 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
On Finite Groups with Element Order Set of Width 3
Eric Min, Chloe So, Yong Yang +3
arXiv21 Oct 2026PreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{hooshmand2026subindices,
  title = {{Subindices of two-element subsets of finite groups}},
  author = {M. H. Hooshmand and Mikhail Kabenyuk},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39463},
  eprint = {2610.39463},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39463},
}

RIS

TY  - UNPB
TI  - Subindices of two-element subsets of finite groups
AU  - M. H. Hooshmand
AU  - Mikhail Kabenyuk
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39463
UR  - https://arxiv.org/abs/2610.39463
AB  - Let A be a nonempty subset of a group G. A right subfactor relative to A is a subset B maximal under inclusion among those for which every element of AB has a unique representation as ab, with a∈ A and b∈ B. Left subfactors are defined analogously using BA. The infima and suprema of the cardinalities of right and left subfactors define four \emph{subindices}. A subset is called \emph{index-stable} if its four subindices coincide. For any infinite group G, every nonempty finite subset of G is index-stable. For finite groups, the situation is more complicated. We prove that for a two-element subset of a finite group, the left and right subindices coincide pairwise. Let A={x,y} with x≠ y, and let X be the cyclic subgroup generated by x^−1y. Then the lower subindex of A is |G:X|·⌈ |X|/3⌉, and the upper subindex is |G:X|·⌊ |X|/2⌋. These formulas imply that A is index-stable if and only if |X| is one of $2,3,4,5,7$. For any two-element subset of a finite group G, the lower subindex is at least ⌈ |G|/3⌉, and the upper subindex is at most ⌊ |G|/2⌋. We establish necessary and sufficient conditions for equality in each of these bounds. All two-element subsets of a finite group are index-stable if and only if every element of the group has order $1$, $2$, $3$, $4$, $5$, or $7$. We give a complete structural classification of such groups.
ER  -

Abstract

Related

Cite