Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 186] THE NOTATION OF HESSE AND CAYLEY

359

uniquely determined by properties A, B, C and that each is
of one of the two types (8), (9).
Using S2, S4, S3, S5, S1 in turn, and property B, we get
S6. 15 48, 26 37, 27 36, 18 45, 14 58, 23 67;
S7. 15 38, 26 47, 27 46, 18 35, 13 58, 24 67;
Ss. 37 47, 38 48, 36 46, 35 45, 13 14, 23 24.
The set S9 with 12 68 has, by S1, S4, the pair 34 57 and
the symbols 16, 26, 47, 37 paired with 28, 18, 35, 45, since not
paired with each other in view of S2. But 16 is not paired
with one of the last three in view of S5. Hence S9 contains the
pair 16 28. A pair 26 35 would imply 47 18 by S7 and hence
37 45, contrary to Ss. A pair 26 45 would imply 37 18 by S6
and hence 47 35, contrary to S7. Hence, by S6,
Ss. 12 68, 34 57, 16 28, 26 18, 37 45, 47 35.
With 12 16 occurs 56 25 by S2, S5, and 68 28 by S9, while
67, 24, 23 are paired with 27, 46, 36, by S1. But 67, 23 are
paired with 27, 36 by S6, and 67, 24 with 27, 46 by S7. Hence
Sio. 12 16, 56 25, 68 28, 67 27, 24 46, 23 36.
By S2, S4, S3 we have the first four pairs of
S11. 26 16, 15 25, 17 27, 28 18, 13 23, 24 14,
and four of the symbols 13, 14, 23, 24, 58, 67. Pairs 58 t, 28 18
would imply 58 18, 28, contrary to S5. Pairs 67t, 17 27 would
imply 67 27, t17, contrary to Slo. Finally, 13 14, 13 24 are
excluded by Ss and S1. Hence we have S11. By the same steps,
S12. 47 16, 38 25, 17 46, 28 35, 58 23, 67 14;
S13. 37 16, 48 25, 17 36, 28 45, 58 24, 67 13.
By S7 6, S, S, S5 or Slo, S1, we get
S14. 26 27, 47 46, 37 36, 16 17, 56 57, 78 68;
S15. 15 18, 38 35, 48 45, 25 28, 56 68, 78 57.
By S1 and S5, the set Si6 with 12 58 has 34 67 and 25 18
and the symbols 15, 47, 37, 28, 46, 36, since 26, 38, 48, 27 are
excluded by Ss, S12, S13, Slo, respectively (since 25 18, 26 imply