THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 225 



formally enunciated. It is of considerable importance in 

 the theory of substitutions, and very easily established. 



Theorem. The transposition of tivo letters in any circular 

 factor always fractures that circle into two: the transposition 

 of two letters of two circular factors always unites those circles 

 into one. 



We have forty-two different substitutions of the fourth 

 order, each of which has four forms, ab=.hc=- cd= da, in terms 

 of the didymous factors ahcd-, and on each (PQ) of these 

 four we form eight irreducible triplets D . PQ, giving in all 

 8 . 42 . 4 irreducible triplets. 



Let D . PQ and D'. PQ be two of the eight irreducibles 

 made on PQ : we cannot have 



D^ Pa=(D . PQ)" = DPQDPa 

 unless 



D'=DPQD = D. (PQ)D-^ 



which is impossible, because D' is of the second order, and 

 DPQD is of the order of PQ, that is, of the fourth. Neither 

 can we have 



D'pa=(DPaD)^=DPaDpaDpa 



unless 



whence 



and 



D' = DPQDPaD, 

 DD'D = PaDPa, 



i = (PQDPQ)% =(DD'D)^=D^; 

 which is false, because PQ is not =(PQ)~^, PQ being of 

 the fourth order. 



It is thus shown that no one of the eight irreducibles 

 made on PQ can be a power of another of them. Hence 

 there are not less than 8r substitutions of the seventh order, 

 no one of which is a power of another ; that is, there are 

 8 . 6r different substitutions of the seventh order ; and as 

 each has four values PQ for the same D, the number 

 8.7.6.4 of triplets irreducible must be divisible by 8 . 6.4r, 

 whence r= i, or r = 7. 



SER. III. VOL. II. Q 



