I am trying search for subgroup lattice diagrame for the general linear group GL$_n(\mathbb F_q)$ but could not find any thing in the net.
Can some one help me by providng some link on it ? thank you
2026-03-31 05:04:51.1774933491
What is subgroup lattice of GL$(n,\mathbb F_q)$?
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Using Gap.app on a Mac (see https://cocoagap.sourceforge.io/ ), or xgap on Linux/Unix (included with the standard gap package distribution, although may require some installing some libraries to build), you can interactively examine the subgroup lattice of any reasonable-sized finite group that you like. E.g.
gap> GraphicSubgroupLattice(GL(3,2));You are unlikely to succeed in viewing the full lattice of a moderate sized $GL_n$. I would suggest looking at limited portions of the lattice, if you would like to gain some intuition. For example, (following the previous GraphicSubgroupLattice command) typing
MaximalSubgroupClassReps(GL(3,2));in GAP, and then doing "InsertVertices from GAP" in the menu of the subgroup lattice. Obviously, replace the maximal subgroups with whatever portion of the lattice you would most like to examine. Working interactively can make it easier to focus on a smallish portion of a large lattice. Right-click on a subgroup to get more information about it.Alternatively, you can get a picture in graphviz format of the subgroup lattice of any group with
DotFileLatticeSubgroups, then use the graphviz program to actually view the diagram. For $GL_n(\mathbb{F}_q)$, best done only with small $n,q$, and interactive work is not (directly?) possible.