336 
[934 
934. 
NOTE ON THE SO-CALLED QUOTIENT U/iY IN THE THEORY 
OF GROUPS. 
[From the American Journal of Mathematics, t. xv. (1893), pp. 387, 388.] 
The notion (see Holder, “ Zur Reduction der algebraischen Gleichungen,” Math. 
Ann., t. xxxiv. (1887), § 4, p. 31) is a very important one, and it is extensively 
made use of in Mr Young’s paper, “On the Determination of Groups whose Order 
is the Power of a Prime,” American Journal of Mathematics, t. xv. (1893), pp. 124— 
178; but it seems to me that the meaning is explained with hardly sufficient 
clearness, and that a more suitable algorithm might be adopted, viz. instead of 
G 1 —G/T 1 , I would rather write G = Tj. QG 1 or QG 1 .Y 1 . 
We are concerned with a group G containing as part of itself a group r t , such 
that each element of r x is commutative with each element of G. This being so, we 
may write 
G = QG 1 .T 1 , 
where QG 1 is not a group but a mere array of elements, viz. if r x = (l, A 2 , ..., A s ), 
and QGi = (l, B 2 , ..., B t ), then the formula is 
G = { 1, B 2 , ..., B t )( 1, A 2 , ..., As), 
where it is to be noticed that the elements B are not determinate; in fact, if A 0 
be any element of r x , we may, in place of an element B, write BA e , for 
B( 1, A 2 , ..., M s ) and BA e ( 1, A 2 , ..., M s ) 
are, in different orders, the same elements of G. 
But, G being a group, the product of any two elements of G is an element of 
G ; viz. we thus have in general 
B{ A. BjAf = B k A k ',