G2 
This corrects the result of my article 96, and it may be of 
interest as affording when generalised, as it easily can he, a 
tactical verification of the very important theorem deduced 
by M. Jordan from a theorem of Galois about a certain 
determinant, that if n r be any power of a prime number n, 
there is a group made with n r letters of 
n(n — 1 — 1 )n\n 3 — 1 ) . . . if (if — 1 ) 
linear substitutions. Vide M. Jordan’s These sur le nombre des 
valeurs des fonctions (Journal de l’Ecole Poly technique, 
1861), chap. v. It also shows that M. Jordan’s group of 
linear substitutions is a maximum modular on the model (j) 
of the order n r , having n r — 1 principal substitutions, i.e. con- 
sisting of (j) and all its derived derangements. 
As this treatise of M. Jordan is the only one, so far as I 
can learn, that has appeared in a revised form in the French 
language since the days of Cauchy, on this difficult subject, 
it may be worth the while to give a little account of it. The 
memoir, as may be expected from the nature of the question, 
is neither short nor simple, but the aim is well defined, and 
the result is easy both to understand and to evaluate. The 
principal object of the work is to determine the order of a 
transitive group T made with M letters, which contains a 
set of similar substitutions PiP 2 P 3 . . ., such that the whole of 
them with their powers and products form an intransitive 
group 2. M. Jordan shows that 2 will contain an intransi- 
M 
tive group made up of -- transitive groups of n letters, and, 
therefore, n being prime, of the nth order ; and if there be 
more than one such group 2 in T, he selects the sous-groupe 
minimum so constructible, and takes its prime order n as the 
modulus of his substitutions. His final result is, that 
M = k'n rp = lcm p , and that the order of T will always of 
necessity divide 
r ’• (H-n * ) k ‘‘ 
l , 2 , 3...&j» 2 •(» — !)(»* — 1)...(ji 1 ' - 1) j ; 
