Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.30952

Orbits of $p$-Nilpotent Group Actions for Small Primes

Chloe So·Yong Yang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $G$ be a finite solvable group and $V$ be a faithful completely reducible $G$-module. We improve previous bounds for the sizes of defect groups of blocks of solvable groups through an improved bound on vector stabilizers of $p$-nilpotent groups. Specifically, if $G$ is 2-nilpotent and $2 \nmid |V|$, then there exists $v \in V$ such that $\mathbf{C}_G(v) \leq \mathbf{F}_4(G)$. Similarly, if $G$ is 3-nilpotent and $3 \nmid |V|$, then there exists $v \in V$ such that $\mathbf{C}_G(v) \leq \mathbf{F}_5(G)$. Using these results, we conclude that $d(B) \leq \lfloor \frac{9n}{10} \rfloor$ when $O_p(G) = 1$ for $p = 2$ and $d(B) \leq \lfloor \frac{19n}{21} \rfloor$ when $O_p(G) = 1$ for $p = 3$.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chloe So, Yong Yang. 2026-10-22. Orbits of $p$-Nilpotent Group Actions for Small Primes. https://arxiv.org/abs/2610.30952

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

KEEP EXPLORING

Related papers

arXiv2026

Random Burnside Groups

Fix integers $m\ge2$ and $k\ge1$. For every sufficiently large fixed odd exponent $M$, choose $k$ independent elements uniformly from the sphere of radius $\ell$ in the free Burnside group $B_m(M)$. We prove that the quotient by their normal closure is infinite with probability at least $1-e^{-c\ell}$. More precisely, the quotient map has no nontrivial kernel element of length at most $κ\ell$, for constants $c,κ>0$. We also consider the analogue of the Gromov positive density model for the Burnside variety.

math.GR↗
arXiv2026

Spherical growth of reciprocal classes in the Hecke Groups

Let $r \in \mathbb{Z}$, $r>2$, and let $Γ_{2r} \cong (\mathbb{Z}/2\mathbb{Z}) \ast (\mathbb{Z}/2r\mathbb{Z})$ be the Hecke group. Let $\mathcal{N}_\ell$ denote the set of conjugacy classes of reciprocal elements with cyclically reduced word length exactly $\ell$ in the standard generating set of $Γ_{2r}$. Let $P(x)$ be the polynomial \[ P(x) = x^{r+1} - 2\sum_{i=1}^{r-1} x^{i} - 1. \] Let $ρ$ be the unique positive root of $P$ and $C$ be a constant depending on $r.$ We prove that, as $\ell \to \infty$, \[|\mathcal{N}_\ell| \sim Cρ^{\left\lfloor \tfrac{\ell+1}{2} \right\rfloor}. \] Our approach exploits the free product structure of $Γ_{2r}$, together with a combinatorial counting function and recurrence relations arising from cyclically reduced representatives. We further show that the asymptotic growth rate of primitive reciprocal conjugacy classes of word length $\ell$ coincides with that of $\mathcal{N}_\ell$. This result extends previous work for odd values of $p$ and provides a uniform asymptotic bound for all Hecke groups.

math.GR↗
arXiv2026

Weak rank rigidity for groups with a navigable path system

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective automorphisms of open convex cones, Helly groups (answering a question of Genevois), Coxeter groups, weak Garside groups (in particular Deligne's groups and fundamental groups of Salvetti complexes of oriented matroids), hierarchically hyperbolic groups, and other examples. Along the way, we show that those groups satisfy the Morse local-to-global property, providing a unified proof for the whole class. In the metric setting, the same condition of non-positive curvature allows to provide a local definition (that is, in a sense, optimal) of rank one/Morse geodesics, mirroring the one using parallel Jacobi fields from Riemannian geometry; to deduce linearity of divergence from linearity on a sequence; to obtain new cases in which Morse geodesics are strongly contracting. The main new tool introduced is the generalised contraction space, a hyperbolic space that encodes the negative curvature of a given space.

math.GR↗