196 Dr. G. A. Miller on the Operation Groups 



The few groups which do not contain such a subgroup will 

 be considered afterwards. 



The eight systems of intransitivity* of the given self- 

 conjugate subgroup are systems of nonprimitivity of the 

 required groups. Hence each one of these groups must have 

 a p, 1 isomorphism to some group of order 8. As all of 

 the latter contain subgroups of order 4, all of the former must 

 contain subgroups of order 4p. 



Since the average number of elements in all the substitu- 

 tions of a group is n — «t> n being the degree and a the num- 

 ber of systems of intransitivity of the group, every subgroup 

 of a regular group must be intransitive ; and an intransitive 

 subgroup of half the order of a transitive group must contain 

 two and only two systems of intransitivity, which are also 

 systems of nonprimitivity of the transitive group. 



Each one of the groups under consideration must therefore 

 contain a subgroup of order 4p, which may be formed by 

 making some regular group of this order simply isomorphic 

 to itself X- Since the groups of order 4p are known §, our 

 problem is reduced to the construction of the nonprimitive 

 groups containing as heads one of the five regular groups of 

 order 4p in 1, 1 correspondence to itself. 



In what follows we shall consider p > 2, as the groups of 

 order 16 are well known ||. We shall first construct all the 

 groups which contain as heads one of the two commutative 

 groups of order 4p in 1, 1 correspondence to itself. The 

 cyclical head will be denoted by H eye, and the non-cyclical 

 by H. 



Groups containing H eye. 



Since there are 2(p — 1) positive integers which are less than 

 Ap and prime to 4p, H eye. contains 2(p — l) substitutions of 

 order 4p. The largest group which transforms H eye. into 

 itself without interchanging its systems transforms these 

 substitutions according to a regular commutative group (L) 



* Every operation group of a given order may be represented by a 

 regular substitution group of the same order. Cf Cayley, ' American 

 Journal of Mathematics/ vol. i. p. 52 ; also Dyck, Mathematische Annalen, 

 vol. xxii. p. 84. 



t Cf. Frobenius, Crelle, ci. p. 287. 



\ Cf. Netto'a ' Theory of Substitutions ' (Cole's translation), § 98. 



§ Cf. Holder, Mathematische Annalen, vol. xliii. p. 411 ; also Cole and 

 Glover, l American Journal of Mathematics,' vol. xv. pp. 202-214. 



|| Young, ' American Journal of Mathematics,' vol. xv. p. 160 ; Holder, 

 Mathematische Annalen, vol. xliii. p. 409 ; Miller, Comptes Hendus, 

 Feb. 17, 1896. 



