Low Diameter Algebraic Graphs
Résumé
Let D be a division ring, n a positive integer, and GLn(D) the set of invertible square matrices of size n and values in D, called the general linear group. We address the intersection graph of subgroups of GLn(D) and prove that it has diameter at most 3. Two particular cases of its induced subgraphs are then investigated: by cyclic subgroups, and by almost subnormal subgroups. We prove that the latter case results in a connected graph whose diameter is sharply bounded by 2. In the former case, we completely characterise the connectivity of the induced graph with respect to D, where, in case of connectivity, we prove that it has diameter at most 7 in general, and at most 5 if D is a locally finite field of characteristic not 2 different from F3 and F9.
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