82 
We have the three equivalent groups 
123456 123456 
461325 4 61325 
354162 354162 
1 2 3 4 5 6 
4 6 1 3 2 5 
3 5 4 1 6 2 
1 2 4 3 6 5 
3 5 1 4 2 6 
4 6 3 1 5 2 
H 
1 5 4 3 2 6 
3 6 1 4 5 2 
4 2 3 1 6 5 
H, 
1 6 4 3 5 2 
3 2 1 4 6 5 
4 5 3 1 2 6 
H 2 
These are equivalent groups, for we have 
IIj— 1 64352H16435 2=P H P" 1 
Hj=l 54326 PI 15432 6=Pj IP Pf 1 
Grouped Groups. 
Def. A principal substitution P of a group G has the form 
f P = A« + B& -f- Cc+ . . jy, 
A7B7 . . 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 
f Q = Ai^i ■+■ + • • 4* Ji/n 
such that the first of the differences 
A — An a — a u B — B l5 b — b u . . . 
which is not zero, is positive. If they be all zero, Q is also 
a principal substitution. 
G6. Theo. H. Let any partition of N be 
N = Aa + B5 4 - Cc + • • 4* 4/ > 
A 7 B 7 C... 7 J 7I. 
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. 
Let PiPz..-p„,q q ..•? r r ...r etc. 
o+l n+2 o-f6 fl+6+1 0+6+2 o+6+c 
be the circular factors of G of the orders ABC,, etc. 
Let p' r be the i th cyclical permutation of p r . 
