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A THEOREM ON GROUPS. 
- 2 VX, 
[From the Mathematische Annalen, t. xm. (1878), pp. 561—565.] 
The following theorem is very simple ; but it seems to belong to a class of 
theorems, the investigation of which is desirable. 
¿i-p, p-e 
symmetrical in 
I consider a substitution-group of a given order upon a given number of letters ; 
and I seek to double the group, that is to derive from it a group of twice the order 
upon twice the number of letters. This can be effected for any group, in a manner 
which is self-evident and in nowise interesting : but in a different manner for a 
commutative group (or group such that any two of its substitutions satisfy the condition 
AB = BA) : it is to be observed that the double group is not in general commutative. 
Let the letters of the original group be abode ..., we may for shortness write 
U =■ abode... \ and take U as the primitive arrangement: and let the group then be 
1, A, B,... where A, B,... represent substitutions: the corresponding arrangements are 
/X + i2, 
U, AU, BU,... and these may for shortness be represented by 1, A, B,...; viz. 
1, A, B, ... represent, properly and in the first instance, substitutions; but when it is 
the first form 
explained that they represent arrangements, then they represent the arrangements 
U, AU, BU,.... 
For the double group the letters are taken to be a^bxdXi ... and a.,bx.d.,e.> .... 
= U 1 and U 2 suppose, and UJJ 2 is regarded as the primitive arrangement; A 1 and A 2 
denote the same substitutions in regard to U x and U 2 respectively, that A denotes in 
regard to U: and so for B 1} B 2 , etc.; moreover 12 denotes the substitution (a x a 2 ) 
(6A2) (CiC 2 ) (dydo) (e x e 2 ) ..., or interchange of the suffixes 1 and 2. The substitutions 
A 1} A 2 , or any powers of these Af, A/, are obviously commutative ; applying them to 
the primitive arrangement U l U 2 , we have A 1 a A.fU 1 U 2 and A/A 1 a U 1 U 2 each = A * U X A£ U 2 . 
But Af, A/ are not commutative with 12: we have for instance 12Af . U X U 2 
= 12A 1 a U 1 . U 2 = AfUn. U 1} but A 1 a 12U 1 U 2 = A x a . U 2 Ux= U 2 . AfU^. If instead of the 
substitutions we consider the arrangements obtained by operating upon UJJ 2 , then we