268 CHAPTER XII. 



all Jf (s)/12 four -groups. Also the 6 four -groups contained in a 

 dihedron 6r 4(r form (under the latter) one system or two systems of 

 conjugate subgroups according as 6 is odd or even, viz., according 

 as s~p n has the form 8ft 3 or Sh 1. Since the 6r 4ff are all 

 conjugate within GM(S), it follows, for 6 odd, that all the four -groups 

 of G M (s) are conjugate; while, for tf even, they form at most two 

 systems of conjugate subgroups under GM(S)- For a even, each 6r 4 

 is one of 0/2 conjugate subgroups of a certain 6r 4(T and is therefore 

 self - conjugate under a subgroup of order 8 of G a . Suppose that, 

 for (? even, the subgroups 6r 4 of GM(S) form a single system of con- 

 jugate subgroups. Then each 6r 4 would be one of M(s)/12 conjugate 

 subgroups and consequently commutative with exactly the 12 operators 

 of a subgroup 6r 12 . By an earlier remark, the G is commutative 

 with a subgroup 6r 8 . Since 8 is not a divisor of 12, our hypothesis 

 is untenable. Hence, for a even, the 6r 4 form exactly two systems 

 of conjugate subgroups of GM( S }- For 1 ) j) > 2, the Jf(s)/12 four- 

 groups 6r 4 contained in GM(*} form one system or two systems of con- 

 jugate subgroups according as s ^p n has the form 8ft + 3 or 8ft 1. 

 In the former case, a 6r 4 is self -conjugate under a 6r 12 ; in the tatter 

 case, under a 6r 24 . In the G^ M (s) the G form a single system of 

 conjugate subgroups and each is self -conjugate under a 6r 24 . Each 6r 12 

 is not a commutative group by 244 and so is of the tetrahedral 

 type ( 247). Likewise, each 6r 24 contains a tetrahedral subgroup 6r 12 . 

 The latter is of index 2 and consequently self -conjugate under 6r 24 . 

 Since 6r 12 contains a set of 4 conjugate 6r 3 , the 6r 24 will contain a 

 complete system of 4 conjugate 6r 3 . Each is self -conjugate under 

 a 6r 6 , which is a dihedron since it is not commutative ( 244). 

 Finally, no operator of period 2 is self -conjugate under 6r 24 ; for it 

 is self -conjugate only under a dihedron G s + i which contains no 

 tetrahedral subgroup and hence none of the present 6r 24 . Then by 

 248 each 6r 24 is an octahedral group. 



247. A non- commutative group of order 12 having a self -conjugate 

 four -group is of the tetrahedral type. 



Let the operators of the four -group be J, F 2 , F 2 ', F 2 ", so that 

 they are commutative and the product of any two F's gives the 

 third F. The 6r 12 contains at least one operator F 3 of period 3. 

 The products . , _ 



*3> > 2 F 3> K 2 K 3> ^2*3 V/ V,L,6) 



are all distinct and so give all the operators of 6r 12 . The 6r 12 would 

 be a commutative group if F 3 were commutative with F 2 , F 2 ', F 2 ". 



1) For p = 2, the four -groups are determined in 249. There are 



i(2_i)( 2 -2) sets. 



