934] NOTE ON THE SO-CALLED QUOTIENT G/H IN THE THEORY OF GROUPS. 337 
that is, 
B{Bj= B k A k 'Aj- 1 Af 1 (i, j, unequal or equal), 
where the B k is a determinate element of the series 1, B 2 , ..., B t , depending only- 
on the elements B { and Bj into the product of which it enters ; and it is in nowise 
affected by the before-mentioned indeterminateness of the elements B: say Bi, Bj 
being any two elements of the series 1, B 2 , ..., B t , we have the last preceding 
equation wherein B k is a determinate element of the same series. 
We may imagine a set of elements 1, B 2 , ..., B t for which, Bi, Bj being any 
two of them and B k a third element determined as above, we have always BiBj = B k , 
that is, these elements 1, B 2 , ..., B t now form a group, say the group G 1 ; the 
original elements 1, B 2) ..., B t (which are subject to a different law of combination 
Bi Bj = B k A Af'Ap 1 , and do not form a group) are regarded as a mere array con 
nected with this group, and so represented as above by QG 1 ; and the relation of 
the original group G to the group Fj (consisting of elements commutative with those 
of G) and to the new group G 1 is expressed as above by the equation 
G = T 1 ,QG 1 , =QG 1 .T 1 . 
Cambridge, 2 June, 1893. 
C. XIII. 
43