If ρ(G) is a finite, real, orthogonal
group of matrices acting on the real vector space V, then
there is defined [5], by the action of
ρ(G), a convex subset of the unit
sphere in V called a fundamental region.
When the unit sphere is covered by the images under
ρ(G) of a fundamental region, we obtain
a semi-regular figure.The group-theoretical problem in this kind of geometry is to find when the
fundamental region is unique. In this paper we examine the subgroups,
ρ(H), of
ρ(G) with a view of finding what
subspace, W of V consists of vectors held
fixed by all the matrices of ρ(H). Any
such subspace lies between two copies of a fundamental region and so
contributes to a boundary of both. If enough of these boundaries might be
found, the fundamental region would be completely described.
Regular orbits of finite primitive solvable groups, III
