The order complex of $PGL_2(p^{2^n})$ is contractible when $p$ is odd

Emilio Pierro

备注：: 3 pages

英文摘要：

Given a group $G$, its lattice of subgroups $\mathcal{L}(G)$ can be viewed as a simplicial complex in a natural way. The inclusion of $1_G, G \in \mathcal{L}(G)$ implies that $\mathcal{L}(G)$ is contractible, and so we study the topology of the order complex $\widehat{\mathcal{L}(G)} := \mathcal{L}(G) \setminus \{1_G,G\}$. In this short note we consider the homotopy type of $\widehat{\mathcal{L}(G)}$ where $G \cong PGL_2(p^{2^n})$, $p \geq 3$, $n \geq 1$ and show that $\widehat{\mathcal{L}(G)}$ is contractible. This is consistent with a conjecture of Shareshian on the homotopy type of order complexes of finite groups.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The covering radius of $\mathrm{PGL}_2(q)$

The covering radius of a subset $C$ of the symmetric group $\mathrm{S}_n$ is the maximal Hamming distance of an element of $\mathrm{S}_n$ from $C$. This note determines the covering radii of the finite projective general linear groups. It turns out that the covering radius of $\mathrm{PGL}_2(q)$ is $q-2$ if $q$ is even, and is $q-3$ if $q$ is odd.

2017-06-02math.CO

On codes in the projective linear group $PGL(2,q)$

In this paper, we resolve a conjecture of Green and Liebeck [Disc. Math., 343 (8):117119, 2019] on codes in $PGL(2,q)$. To be specific, we show that: if $D$ is a dihedral subgroup of order $2(q+1)$ in $G=PGL(2,q)$, and $A=\{g\in G: g^{q+1}= 1,\, g^2\ne 1 \}$, then $λG=A\cdot D$, where $λ=q$ or $q-1$ according as $q$ is even or odd.

2020-09-02math.CO

Nets of conics of rank one in PG(2,q), q odd

We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent problem of classifying the orbits of planes in $\text{PG}(5,q)$ which meet the quadric Veronesean in at least one point, under the action of $\text{PGL}(3,q) \leqslant \text{PGL}(6,q)$ (for $q$ odd). Our results complete a partial classification of nets of conics of rank one obtained by A. H. Wilson in the article "The canonical types of nets of modular conics",…

2020-03-13math.AGmath.CO

Pentavalent symmetric graphs of order four times an odd square-free integer

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of order $4pq$. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\itΓ$, either the full automorphism group ${\sf Aut}\itΓ$ is isomorphic to ${\sf PSL}(2,p)$, ${\sf PGL}(2,p)$, ${\sf PSL}(2,p){\times}\mat…

2017-02-19math.CO

Fissioned triangular schemes via sharply 3-transitive groups

n [D. de Caen, E.R. van Dam. Fissioned triangular schemes via the cross-ratio, {Europ. J. Combin.}, 22 (2001) 297-301], de Caen and van Dam constructed a fission scheme $\FT(q+1)$ of the triangular scheme on $\PG(1,q)$. This fission scheme comes from the naturally induced action of $\PGL(2,q)$ on the 2-element subsets of $\PG(1,q)$. The group $\PGL(2,q)$ is one of two infinite families of finite sharply 3-transitive groups. The other such family $\Mq(q)$ is a "twisted" version of $\PGL(2,q)$, where $q$ is an even power of an odd prime. The group $\PSL(2,q)$ is the intersection of $\PGL(2,q)$ a…

2011-07-02math.COmath.GR

On pencils of cubics on the projective line over finite fields of characteristic $>3$

In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by 3. These equivalence classes are considered as orbits of lines in $\mathrm{PG}(3,q)$, under the action of the subgroup $G\cong \mathrm{PGL}(2,q)$ of $\mathrm{PGL}(4,q)$ which preserves the twisted cubic $\mathcal{C}$ in $\mathrm{PG}(3,q)$. In particular we determine the point orbit distributions and plane orbit distributions of all $G$-orbits of lines which are contained in an osculating plane of $\mathcal{C}$, have non-empty intersection w…

2021-04-10math.AGmath.CO

更多相关论文收起更多论文

The homotopy type of the independence complex of graphs with no induced cycles of length divisible by $3$

We prove Engström's conjecture that the independence complex of graphs with no induced cycle of length divisible by $3$ is either contractible or homotopy equivalent to a sphere. Our result strengthens a result by Zhang and Wu, verifying a conjecture of Kalai and Meshulam which states that the total Betti number of the independence complex of such a graph is at most $1$. A weaker conjecture was proved earlier by Chudnovsky, Scott, Seymour, and Spirkl, who showed that in such a graph, the number of independent sets of even size minus the number of independent sets of odd size has values $0$, $1…

2022-03-08math.ATmath.CO

Terraces for Small Groups

We use heuristic algorithms to find terraces for small groups. We show that Bailey's Conjecture (that all groups other than the non-cyclic elementary abelian 2-groups are terraced) holds up to order 511, except possibly at orders 256 and 384. We also show that Keedwell's Conjecture (that all non-abelian groups of order at least 10 are sequenceable) holds up to order 255, and for the groups $A_6$, $S_6$, $PSL(2,q_1)$ and $PGL(2, q_2)$ where $q_1$ and $q_2$ are prime powers with $3 \leq q_1 \leq 11$ and $3 \leq q_2 \leq 8$. A sequencing for a group of a given order implies the existence of a com…

2016-03-04math.COmath.GR