By A B Sosinskiĭ

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Is such a situation possible? Of course it is, but only if G+ consists of rotations about the unique axis x1 x2 . But then it follows that G+ ∼ = Zn for some n ≥ 2. So the theorem is proved for the case |F | = 2. Note that in this case v(x1 ) = v(x2 ) = n = |G+ |. It is easy to see that if the action of G+ on F produces only two orbits, then the stabilizers of points from these two orbits have the same number of elements and we are in the case |F | = 2 considered above. Thus for the rest of the proof, we can assume that there are three orbits.

In this section we study the Coxeter geometries in R3 . , the bounded intersection of a finite number of half-spaces in R3 ) with dihedral angles of the form π/k for various values of k = 2, 3, . . 3. 3). 2. Theorem. 3. It is not very difficult to prove that the seven polyhedra (listed in the theorem) indeed define Coxeter geometries. To prove that there are no other geometries, nontrivial information from linear algebra (in particular, the notion of Gramm matrix) is needed. 3, no. 7, 2003). A remark about terminology.

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