82 



We have the three equivalent groups 



123456 123456 123456 



4 61325 4 6132 5 4 61325 



35416 2 3 54162 3 54162 



124 3 65 15 432 6 164352 



351426 361452 321465 



4 6 3 15 2 4 2 3 16 5 4 5 3 12 6 

 H Hi Ho 



These are equivalent groups, for we have 



Hi=l 64352H16435 2=P H P"^ 



H2=nl 5 4 3 2 6 H 1 5 4 3 2 6c=Pi H Pf'- 



Grouped Groups. 

 Def. A principal substitution P of a group G has the form 

 /P = Aa + B5 -I- Cc+ . . J/, 

 A-7B7 . .7 J, 

 which means that P has a circular factors of the order A, 

 b of the order B, etc. Every other substitution Q of the 

 group has the form 



/Qr:zA,«i + BA + .. + Ji;;^ 



such that the first of the differences 



A — Ai, a — Gi, B — Bi, 5 — ^1, . . . 

 which is not zero, is positive. If they be all zero, Q is also 

 a principal substitution. 



C6. Theo. H. Let any partition of N be 

 N = A« -f BS + Cc + . . -f Ji , 

 A7B7C. . .7Jy 1. 

 K being the least common multiple of A B C , . J, and such 

 that one at least of a b . ,j is 7 1. 



Let G be any one of the W groups of K powers of a sub- 

 stitution constructible on this partition of N, by Theorem C, 

 Letpip. . . .i?«, g q ' ' -q r r , , ,r etc. 



a+l rt+2 a-\-b «+&+! rt+J+3 «+6+c 



be the circular factors of G of the orders ABC,', etc. 

 Let pi be the i^^ cyclical permutation of /?^. 



