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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers in keeping with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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120/2 = 7140 pairs of tritangent planes of the general curve. 7140/3 = 152320 even triads of tritangent planes of the general curve. 8 (the last column of page 414). 240/3 = 2240, since those at any vertex correspond to the diameters of (PA)1. 240/24 = 3150, since those at any vertex correspond to the diagonal cubes of (PA)7 (or to the pairs of opposite diagonal tetrahedra). , that lying in the 4-space X^-{-X2 = #3"T#4 — ^ 5 ~ r ^ 6 ' X7 = X8 = X§, 2J X = 0) we can inscribe three y4's or three /34 -\/2's, making 9450 of each altogether.

Of ppn, is . e. of nnn, conas a sub-group of index 2, and is itself a sub-group of index 3 tains i on in the group of symmetries of Onnn. r3nBy (12 . 51) is always (»+l)! (p+l)\ (q+l)\ [npq]. 6. group 32J The order of the unextended is just half as great. J (# + 2 + 2)1 2"+2(n + 3)! J i(P + 2«-2)l 2"+1(« + 3)! 25920 1451520 348364800 160 H. S. M. 1, 3,3 is the group of auto"3, 3"1' morphisms of the 27 lines on a general cubic surface. 3, j self-conjugate sub-group of index 2, must therefore be identical with the simple group* A (4, 3).

The most important of our extended groups is 3,3" 3,3 3 since this, being the group of symmetries of 221 or (PA)6, is also the group of automorphisms of the lines on a general cubic surface. By (16 . <})* =(QN)*=(QN1)* 2 = (JVP) = (Nx P) 2 = (NPJ2 = (Nx P x ) 2 = 1. For simplicity we have written N for No, P for P o , Q for Qo. 164 H\ S. M. COXETER [March 12, This abstract definition is conveniently represented by the following diagram: N P Q The six generating operations are each of period 2; all pairs of them not directly linked in the diagram are permutable, and the products of linked pairs are of period 3.