A characterisation of weakly locally projective amalgams related to A₁₆ and the sporadic simple groups M₂₄ and He

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• 作者：Michael Giudici^a ; ^{michael.giudici@uwa.edu.au" class="auth_mail" title="E-mail the corresponding author} ; A.A. Ivanov^b ; ^{a.ivanov@imperial.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Luke Morgan^a ; ^{luke.morgan@uwa.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Cheryl E. Praeger^a ; ^{cheryl.praeger@uwa.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20B25 ; 05C25 ; 20D08
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 August 2016
• 年：2016
• 卷：460
• 期：Complete
• 页码：340-365
• 全文大小：501 K

文摘

A simple undirected graph is weakly G-locally projective, for a group of automorphisms G, if for each vertex x  , the stabiliser G(x)$G(x)$ induces on the set of vertices adjacent to x   a doubly transitive action with socle the projective group L_nx(q_x)$L_{n_x}(q_x)$ for an integer n_x$n_x$ and a prime power q_x$q_x$. It is G-locally projective if in addition G is vertex transitive. A theorem of Trofimov reduces the classification of the G  -locally projective graphs to the 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 weakly locally projective graphs, we would like to begin a program of characterising some of the remarkable examples. We show that if a graph is weakly locally projective with each q_x=2$q_x=2$ and n_x=2$n_x=2$ or 3, and if the distance factors are as in the examples arising from the rank 3 tilde geometries of the groups M₂₄$M_{24}$ and He  , then up to isomorphism there are exactly two possible amalgams. Moreover, we consider an infinite family of amalgams of type U_n${\mathcal{U}}_n$ (where each q_x=2$q_x=2$ and n=n_x+1≥4$n=n_x+1\ge 4$) and prove that if n≥5$n\ge 5$ there is a unique amalgam of type U_n${\mathcal{U}}_n$ and it is unfaithful, whereas if n=4$n=4$ then there are exactly four amalgams of type U₄${\mathcal{U}}_4$, precisely two of which are faithful, namely the ones related to M₂₄$M_{24}$ and He  , and one other which has faithful completion A₁₆$A_{16}$.