English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A simple undirected graph is weakly $G$-locally projective, for a group ofautomorphisms $G$, if for each vertex $x$, the stabiliser $G(x)$ induces on theset of vertices adjacent to $x$ a doubly transitive action with socle theprojective group $L_{n_x}(q_x)$ for an integer $n_x$ and a prime power $q_x$.It is $G$-locally projective if in addition $G$ is vertex transitive. A theoremof Trofimov reduces the classification of the $G$-locally projective graphs tothe case where the distance factors are as in one of the known examples.Although an analogue of Trofimov's result is not yet available for weaklylocally projective graphs, we would like to begin a program of characterisingsome of the remarkable examples. We show that if a graph is weakly locallyprojective with each $q_x =2$ and $n_x = 2$ or $3$, and if the distance factorsare as in the examples arising from the rank 3 tilde geometries of the groups$M_{24}$ and $He$, then up to isomorphism there are exactly two possibleamalgams. Moreover, we consider an infinite family of amalgams of type$\mathcal{U}_n$ (where each $q_x=2$ and $n=n_x+1\geq 4$) and prove that if$n\geq 5$ there is a unique amalgam of type $\mathcal{U}_n$ and it isunfaithful, whereas if $n=4$ then there are exactly four amalgams of type$\mathcal{U}_4$, precisely two of which are faithful, namely the ones relatedto $M_{24}$ and $He$, and one other which has faithful completion $A_{16}$.

A characterisation of weakly locally projective amalgams related to $A_{16}$ and the sporadic simple groups $M_{24}$ and $He$