46 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
is commutative with each of the 2m. For if (a) one of the 2m is non-commutative 
with (v), the chain N contains the pair a, and thus (a) is non-commutative with (N) 
also, and therefore commutative with (Nv) ; and if (a) is commutative with (v ), N 
does not contain a, and thus (a) is commutative with (N) also, and therefore also 
with (NF). 
267. Suppose now we have any pure complete network consisting of 2m networks 
such as the 2 m of the preceding sections, and also of other' simple networks (X), (g), 
(v)... ; we can substitute for the latter the simple networks (LX), (Mg), (Nv) . . . where 
the (L), (M), (N). . . are networks such as the (N) of the last section, and we shall still 
have a pure complete network, and it will be such that the networks other than the 
2m are commutative with the 2m. Now these networks other than the 2m are either 
all commutative with each other, or else there are two at least which are not; in the 
latter case we may add such two to the 2m and get 2m+2 such as the 2m; we may 
then as before substitute for the remainder simple networks which are all commutative 
with the 2m+2, and may repeat the process continually until the remaining networks 
other than the conjugate pahs are all commutative with each other. 
268. Any group of the family we are considering is such that all the simple 
networks are commutative, or that two at least are not; in the latter case, if we take 
the two as conjugates, and take in others so as to constitute a pure complete network, 
we can proceed to deal with this in the mode we have just indicated. Thus in every 
case we can obtain a pure complete network containing n lots of two conjugate 
networks, and one lot of r commutative networks, the networks of each of the 
(n-j-r) lots being commutative with those of the others. Either n or r may vanish. 
269. If we make n — 4 and r— 1 we have a group of 16 units given by the following 
table :— 
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u 
