152 
A THEOREM ON GROUPS. 
[659 
As the most simple instance of the theorem, suppose that the original group is 
the group 1, (abc), (acb), or say 1, ©, © 2 , of the cyclical substitutions upon the 3 letters 
abc. Here m 2 = 1 (mod. 3) or except m = 1 the only solution is m = 2, and thence 
A=l. The double group is a group of the order 6 on the letters a^c^b.x.,: viz. 
writing © = (a&c), and therefore ©i = (ui&iCi), ©i 2 = (a^b^, © 2 = (a 2 6 2 c 2 ), ©v* = (a 2 c 2 6 2 ), also 
writing 12 = a, the substitutions are 
1, ©J02 2 , ©i 2 ©.,, 
a, a©!©./, a©! 2 ©^ 
the arrangements corresponding to the second row of substitutions are a^Kc.^a-J)^, 
6 2 c 2 a 2 c 1 a 1 6 1 , c. 2 a 2 b 2 b 1 c 1 a 1 , viz. the substitutions are {a 2 a. 2 ) (&A)(CiC 2 ), (aA) (&iC 2 ) (Cia 2 ), 
(cqc,) (Ms) (cA), each of them of the second order as they should be. 
I take the opportunity of mentioning a further theorem. Let g be the order of 
the group, and a the order of any term A thereof, a being of course a submultiple 
of g: and let the term A be called quasi-positive when g ^1 — is even, quasi 
negative when g ^1 — -J is odd. The theorem is that the product of two quasi 
positive terms, or of two quasi-negative terms, is quasi-positive; but the product of a 
quasi-positive term and a quasi-negative term is quasi-negative. And it follows hence 
that, either the terms of a group are all quasi-positive, or else one half of them are 
quasi-positive and the other half of them are quasi-negative. 
The proof is very simple: a term A of the group operating on the g terms 
(1, A, B, G,...) of the group, gives these same terms in a different order, or it may 
be regarded as a substitution upon the g symbols 1, A, B, G, ...; so regarded it is 
a regular substitution (this is a fundamental theorem, which I do not stop to prove), 
and hence since it must be of the order a it is a substitution composed of - cycles, 
each of a letters. But in general a substitution is positive or negative according as 
it is equivalent to an even or an odd number of inversions; a cyclical substitution 
upon a letters is positive or negative according as a — 1 is even or odd; and the 
substitution composed of the - cycles is positive or negative according as - (a — 1), 
ci a 
that is, /X ^1 — ^, is even or odd. 
Hence by the foregoing definition, the term A, 
according as it is quasi-positive or quasi-negative, corresponds to a positive substitution 
or to a negative substitution; and such terms combine together in like manner with 
positive and negative substitutions. 
Cambridge, 3rd April, 1878.