GROUPS AND Many-valued functions. 277 



group. We may write the circular factors under the first 

 of their elements which occurs in the substitution thus : 

 312564, 465132 



2 6 12 3 



1 4 



358 1 24066^907 • 



8 2 c a 



6 7 9 



4 

 1 • 



In the notation of Cauchy the last substitution is repre- 

 sented thus : 



(1 3 8 6 4) (7 o c) (9 « b) (2 5), 

 of which the inconvenience is that we lose sight of the 

 form which the substitution wears, viz., 

 358i24o66c9a7, 

 in a model group. 



5. The product of two substitutions PQ varies in general 

 with the written order of the two factors. But in some 

 cases we have 



pa=QP. 



When this is true we say that Q is permutable with P. 



If we change in any way the sequence of the circular 

 factors of the same order, and write the changed sequence 

 over the given one, we have always a substitution permu- 

 table with the given one. Thus let the given one be 

 39682iao547c b=V 



6 5 7 6 



12 4 



made with thirteen elements. 



We can form Q, and Q,', both permutable with P, thus : 



~~^L^—^=A. 28 1 ';o7396ac6=Q, 

 361-804 J ! ov 



§5^1136^:^^=428 i5o6396c7«=a' 



36l.804-a7.c6 J ov / 



and we have 



QP=PQ 



QT=PQ', 



as it may be proved by the rule given in (3). 



This is a theorem of Cauchy's, whose somewhat difficult 



