By H.S.M. and W.O.J. Moser Coxeter
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In the main diagonal we give the (smallest) index of each group as a proper subgroup of itself. The details of the work are too lengthy to be given here. (They may be seen in MoSER's Ph. D. ) However, it seems worth while to show how the two most complicated groups, p 4 m and p 6 m, together contain all the others as subgroups. 511, R1 and S = RR 2 generate p 4 g"' [4+, 4], Sand T1 = R 1 R 2 generate p 4 "' [4, 4+], R, R' = R 2 S and T 1 generate c m m, P = R 1 S and 0 = S R 1 generate p g g, 52 5ol Regular tessellations R andP PandQ generate c m, = RPR generate p g, R 4 = RR1 R, Tv and T 2 =5 2 generate p m g, Tl> T 2 , and T 3 = RT1 R generate p 2, R1 , R 2 , R3 = SR and R4 generate p m m ~ [oo] X [oo], R 1 , R 3 , and Y = R 2 R4 generate pro"' [oo]X[oo]+, X= R 1 R 3 andY generate p 1"' [oo]+x [oo]+o In p 6 m"' [3, 6], defined by 40517, Rand 5 = R1 R2 generate p 31m"' [3+, 6], 5 and T= R 2 R generate p 6"' [3, 6]+, R1 , R 2 and R 3 = RR1 R generate p 3m 1 51 = R1 R 2 and 5 2 = R 2 R3 '"'-J t-,, generate p 3 "' 6 + o In the last column of Table 3, the symbols for the generators have been slightly altered (for simplicity, and to stress certain analogies) o Chapter 5 Hyperbolic Tessellations and Fundamental Groups After describing the regular tessellations {p, q} (which are simplyconnected topological polyhedra), we shall derive the fundamental groups (§ 305, po 24) for closed surfaces of arbitrary characteristic.
85). The above surfaces are orientable. The simplest non-orientable surface is the projective plane, which may be regarded as a sphere with antipodes identified, or as a hemisphere with opposite peripheral points identified, or as a sphere with a cross-cap (HILBERT and CoH~-VosSEN 1932, p. 279). Its fundamental group is the ~ 2 defined by (LEFSCHETZ A 2 =E, whose single generator may be identified with a straight line of the projective plane. 53) Ai A~= E, which is infinite since it has a factor group ;t)= defined by Ai =A~= E.
Since this group is the direct product of the free groups {X} and {Y}, it may be described as ~ ... X~oo· Its general elementX~Y" (where x and y are integers) is the translation from the origin to the point with affine coordinates (x, y). t be convex, it must have a centre of symmetry (FEDOROV 1885, pp. 271-272) and in fact must be either a parallelogram (as above) or a centrally symmetrical hexagon. 5011) XYZ=ZYX=E (Fig. 5b). By changing the magnitudes and directions of the translations X, Y, we obtain infinitely many geometrical varieties of the single abstract group.