Subgroups of \(\text{GL}_n(q)\) which fix sublattices of the subspace lattice
Let \(n\) be a positive integer and let \(q=p^e\) be a prime power. An interesting problem of computational group theory is that of producing an algorithm which, given a collection of nonsingular \((n\times n)\)-matrices over the field \(\text{GF}(q)\), returns a useful description of the subgroup of \(\text{GL}_n(q)\) which they generate. A key tool in the algorithms developed so far is Aschbacher's description of the maximal subgroups of \(\text{GL}_n(q)\). That is, \textit{M. Aschbacher} [in Invent. Math. 76, 459-514 (1984; Zbl 0537.20023)] described nine classes \({\mathcal C}_1,\dots,{\mathcal C}_8\), \(\mathcal S\) of proper subgroups of \(\text{GL}_n(q)\) and showed that every proper subgroup of \(\text{GL}_n(q)\) is contained in a group from these nine classes. The subgroups of \(\text{GL}_n(q)\) from classes \({\mathcal C}_1,\dots,{\mathcal C}_8\) are (for the most part) stabilizers in \(\text{GL}_n(q)\) of ``natural'' algebraic structures on the corresponding vector space. If a subgroup \(H\) of \(\text{GL}_n(q)\) is not contained in any member of \(\bigcup_{i=1}^8{\mathcal C}_i\) then \(H/Z(H)\) is almost simple. The group \(G=\text{GL}_n(q)\) acts on the lattice \(B_n(q)\) of all subspaces of the \(n\)-dimensional space \(V=V(n,q)\) over \(\text{GF}(q)\) on which \(G\) acts naturally. For a sublattice \(\mathcal L\) of \(B_n(q)\), by \(G_{\mathcal L}\) is denoted the setwise stabilizer of \(\mathcal L\) in \(G\). Members of certain classes \({\mathcal C}_i\) are stabilizers in \(G\) of certain sublattices of \(B_n(q)\) and the \({\mathcal C}_i\), whose elements are sublattice stabilizers, are known. Recent algorithms (see \textit{C. R. Leedham-Green} and \textit{E. A. O'Brien} [J. Algebra 189, No. 2, 514-528 (1997; Zbl 0884.51006), Int. J. Algebra Comput. 7, No. 5, 541-559 (1997; Zbl 0907.20025)]) involve recognizing \(H\) as being contained in a member of \({\mathcal C}_i\) for certain \(i\) by showing that \(H\) stabilizes some sublattice of \(B_n(q)\). The main theorem of the present paper states the following: if \(\mathcal L\) is a nontrivial proper sublattice of \(B_n(q)\) (i.e., \({\mathcal L}\not=B_n(q)\) and \(\mathcal L\) contains a nontrivial proper subspace of \(V\)) then either \(G_{\mathcal L}\) is contained in an element of \({\mathcal C}_i\) for some \(i\in\{1,2,3,4,5,7\}\), or every nontrivial proper subspace of \(V\) contained in \(\mathcal L\) has dimension \(n/2\) and any two such subspaces intersect trivially. Furthermore, the authors provide some examples of elements of \(\mathcal S\) which stabilize nontrivial sublattices of \(B_n(q)\).
scientific article; zbMATH DE number 1891316