70 G. A. Miller. 



by qr, when p — 1 is divisible by q but not by r thei*e is only 

 one such group, when p — 1 is not divisible by q there is no 

 group of this type. 



The smallest set of values of p, q, r is 5, 3, 2. Since 5 — 1 = 4 

 is not divisible by 3 there is no transitive group of degree 10 and 

 Order 30 ^). The second smallest set of values is 7, 3, 2. In this 

 case 7^1 = 6 is divisible by both, 3 und 2. Hence there are 

 two transitive groups of degree 14 and order 42. These two 

 groups contain the following substitutions. 



abcdefg . hijJdmn ahcdefg . hijJdnin 



aceghäf . hjlnikm aceghdf . hjlnikm 



adgcfbe . hkujmil adgcfbe . liknjmil 



aebfcgd . hlimjnk aehfcgd . liUmjnk 



afdhgec . limkinlj afdbgec . limkinlj 



agfedcb . hnmlkji agfedcb . Immlkji 



bce . dgf . iß . kmn bc.e . dgf . ijl . knm 



abd , cfe . hik .jml abd . cfe . hik .jml 



acg . bed . hjn . ilk acg . bed . lijn . ilk 



ade . bfg . likj . imn ade . bfg . Jikj . imn 



aef . bge . hlin . inj aef . bge . Mm . inj 



afb . deg . limi . kln afb . deg . hmi . kln 



age . cdf . Iml .jkm age . cdf . Jini .jkm 



bee . dfg . ilj . kmn bee . dfg . ilj . kmn 



adb . eef. hki .jlm adb . eef. liki .jlm 



age . bde . hnj . ikl age . bde . Imj . ikl 



aed . bgf . hjk . imn aed . bgf . hjk . inm 



afe . beg . Jiml . ijn afe . beg . liml . ijn 



abf . dge . liim . knl abf . dge . liim . knl 



aeg . rfd . Idn .jmk aeg . cfd . hin .jmk 



all . bi . cj . dk . el .fm . gn all . bn . cm . dl . ek .fj . gi 



aickemghbjdlfn ai . bli . en . dm . el .fk . gj 



ajenbkfliclgidm aj . bi . eh . dn . em .fl . gk 



akgjfielidnembl ak . bj . ei . dh . en .fm . gl 



albmcndheifjgk al . bk . cj . cli . eh .fn . gm 



^) Cf. Cole's enumeration of the transitive groups of degree 10 in Quarterly 

 Journal of Mathematics, vol. 27, pp. 40 — 44. 



