64 
groupe minimum, M = 4 , 3 1 ' 1 = kn n ‘, and the order of T should 
divide 
l-2-3-4{3-2} 4 , 
but 60' 120 is no divisor of the number. In like manner we 
can construct transitive groups T of the orders, 120'240, 
360-720, and 720' 1440. 
We thus easily show that M. Jordan’s limit is defective 
for such transitive groups T made with 2 n r = M letters, (Jc - 2) 
whatever n r may be > 4 ; and also for higher values of Jc. 
All the groups T considered by M. Jordan are portions of 
the transitive maximum woven grouped group of my 
theorem L made with 
kn rp = n r l = A a 
elements, of which the order is 
(» r -(» r - l)...3-2-l)'l-2-3../. 
When M=2-3 i :i =^-^, or = 3-2 ™ = £•«*, or = 2*2 21 = 
M. Jordan’s limit coincides with mine. The tactical con- 
struction of this group is very simple, and requires no aid of 
analytic formulae, or numerical computation. It is very 
desirable that this maximum group should be broken into its 
factors, but the key to this division is yet to be found. I shall 
show, that the required division can be in a great measure 
effected by selecting the positive groups and the mixed 
groups composing it. * 
If we treat M. Jordan’s group of linear substitutions made 
with n r elements, n being prime, by the method of my article 
(84) ( Theory of Groups), we easily establish the theorem 
following, which is probably new : — 
Theorem. — If n r be any power >2 of a prime number n >2, 
there is a transitive non-modular group formed with n r - 1 
elements of (n r - 1 )w r ~ 1 . {n r ~ l - 1 ) n r ~ 2 . ( n r ~ 2 - 1 ) . . . » 2, (n 2 - 1 ) n (n - 1 ) 
substitutions ; also a transitive group made with n r ~2 elements 
of n r ~ 1 .(n r ~'-l)n r -.(n r ~ 2 . (n r ~ 2 *l)... n 3 (n 2 -l)n(n- 1) substitu- 
tions, which is generally non-modular. 
