Description of D-complete subgroups in the general linear group over the field of three elements

Abstract

If ¦K¦ ⩾ 4, then in GL(n, K) over the field K the subgroup of diagonal matrices is pronormal and hence the lattice of intermediate subgroups admits a description in terms of the concept of a fan (RZhMat, 1980, 5A207). Now if ¦K¦=3, then the subgroup of diagonal matrices D is not pronormal and not even fanlike in the general linear group G, and in this case the lattice ℒ of intermediate subgroups has a more complicated structure. The author, as the first step in the study of ℒ, has obtained a description of all D-complete subgroups in G, i.e., those intermediate F for which DF=F.