Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.20516

On a class of combinatorial group invariants

Arthur Fernandes·Claudio Qureshi·Lucas Reis·Sávio Ribas
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we explore group invariants arising from combinatorial structures associated with finite groups, including the functional graphs of power maps and the well-studied power graphs. These invariants induce equivalence relations (and hence partitions) on the set of isomorphism classes of finite groups, which we classify from the finest to the coarsest. Surprisingly, all but three of these partitions turn out to coincide; for the subclass of nilpotent groups, all but two coincide. Furthermore, we introduce a broad class of nilpotent groups and show that, within this class, all but one of these partitions agree. Our proofs draw on tools and ideas from Combinatorics and Number Theory, while requiring only basic notions from Group Theory. In particular, we construct a general framework that may prove useful in contexts similar to those considered in this paper. Finally, we propose some open questions that emerge from our results.

Explore related subjects

math.COmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Arthur Fernandes, Claudio Qureshi, Lucas Reis, Sávio Ribas. 2026-09-17. On a class of combinatorial group invariants. https://arxiv.org/abs/2609.20516

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

KEEP EXPLORING

Related papers

arXiv2026

Orthogonal Pairs in Maps from the Sphere to the Circle

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $π/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

math.CO↗
arXiv2026

Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, Δ(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $Δ(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $Δ(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $Δ(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $Δ(9)$ remains open.

math.CO↗
arXiv2026

Chromatic symmetric functions for annular webs

We introduce a combinatorial definition of chromatic symmetric functions for annular webs. We prove their symmetry by constructing a web analogue of the Shareshian--Wachs involution and show that they coincide with the symmetric functions associated to annular webs via Turaev's isomorphism. We then derive explicit formulas for their hook Schur coefficients. We also introduce web LLT functions, whose hook Schur coefficients admit positive Laurent-polynomial formulas. These formulas yield a combinatorial expression for the coefficients of the HOMFLY--PT polynomial of an annular web.

math.CO↗