Automorphism group of a local inclusion graph over the general linear group.

AbstractThe inclusion graph of a group <italic>G</italic>, written as In(G), is defined to be an undirected graph with all nontrivial subgroups of <italic>G</italic> as its vertices, and two distinct subgroups <italic>H</italic>, <italic>K</italic> of <italic>G</italic> are adjacent if either H≤K or K≤H. A local inclusion graph of <italic>G</italic> with respect to a given subgroup <italic>S</italic>, written as In(G,S), is an induced subgraph of In(G) with vertex set consisting of all nontrivial subgroups of <italic>G</italic> properly containing <italic>S</italic>. Indeed, In(G) can be viewed as the special case when <italic>S</italic> is the identity subgroup. Let GLn(F) be the general linear group consisting of all n×n invertible matrices over a field F and Tn(F) the subgroup of all invertible lower triangular matrices in GLn(F). In this article, an arbitrary automorphism of In(GLn(F),Tn(F)) is determined and it is proved that the automorphism group of In(GLn(F),Tn(F)) is isomorphic to the direct product of Sn−1 and S2, where Sn−1 is the symmetric group of degree n−1. [ABSTRACT FROM AUTHOR]