Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.17583

On the number of orthomorphisms of the alternating group on four symbols

Vivek Kumar Jain·Rohitesh Pradhan
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this study, the set of orthomorphisms of a finite group G of order 3M with a normal subgroup of order M has been partitioned into classes. Additionally, a number of relationships between the orders of these classes have been established. Ultimately, this approach has been used to theoretically determine all 3840 orthomorphisms of alternating groups on four symbols by just counting 30 orthomorphisms, a problem that had been unsolved since 1992.

Explore related subjects

math.GR

Keep this discovery

BibTeXRIS

Vivek Kumar Jain, Rohitesh Pradhan. 2026-06-16. On the number of orthomorphisms of the alternating group on four symbols. https://arxiv.org/abs/2606.17583

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

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗