Vol. 7, 1921 
MATHEMATICS: G. A. MILLER 
147 
Finite Order" 1897 and 1911. The system of groups in question consists 
of q—1 distinct groups for all values of p and q which satisfy certain con- 
ditions to be noted later while only one such group is given in the pub- 
lished lists. This system of groups is also interesting in view of the fact 
that each group of the system contains q characteristic operators includ- 
ing the identity. 
To construct the groups of the system in question suppose that q is 
a divisor of p—1 and establish a simple isomorphism between q cyclic 
groups of order pq written as regular substitution groups. Let t repre- 
sent the substitution of order q which permutes the corresponding letters 
of these q cyclic groups so that t is commutative with each of its substi- 
tutions and together with the cyclic group of order pq formed by the given 
isomorphism generates a regular abelian group of order pq 2 . Let Sj S 2 
S Q be substitutions of order q and of degree pq — q which transform corre- 
sponding generators of the given cyclic groups of order pq into the same 
power belonging to exponent q modulo pq and so chosen that the product 
5/ S 2 ..S Q is commutative with t. Finally let So represent a substitution 
of order q contained in the first one of the q given cyclic groups of order pq. 
The product SoSjS 2 ..S q t is a substitution of order q 2 whose q th power 
is the substitution of order q, in the group formed by means of the said 
isomorphism, whose constituent is So. Hence it results that the pq 
substitutions of the group of order pq 2 thus constructed which transform 
into a given power belonging to exponent q modulo pq a generator of the 
given cyclic group of order pq can be so chosen that their q th power is 
an arbitrary operator of order q contained in this cyclic group. The 
totality of these pq substitutions must correspond to itself in every auto- 
morphism of this group of order pq 2 . Hence this q th power must be a 
characteristic operator of the group. 
From the preceding paragraph it results that each of the groups of 
order pq 2 under consideration contains q characteristic operators in- 
cluding the identity and that any of these operators which is of order q 
can be made the q th power of all the pq operators of the group which 
transform the operators of order p in the group into a particular power. 
Hence there are q—1 distinct groups for particular values of p andq which 
satisfy the conditions that p and q are such primes that p— 1 is divisible 
by q. These q—1 groups are conf ormal : that is, they contain the same 
number operators of each order. It is well known that for any prime 
number of q there is an infinite number of prime numbers p such that 
^> — 1 is divisible by q and hence there is no upper limit to the number of 
such distinct conf ormal groups. 
The smallest order for which there exist at least two such conformal 
groups is 63. In this special case one of the characteristic operators 
of order 3 is the third power of the operators of order 9 which transform 
the operators of order 7 into their fourth powers while the other operator 
of order 3 is the third power of those which transform the operators of 
