242 
MATHEMATICS: G. A. MILLER 
is that these two subsets have no common operator. If two of the 
operators ^i, 52, . . ., are of different orders, the larger operator is 
transformed into a power of itself by the smaller, since these two oper- 
ators generate a group of order pq, p and q being distinct prime numbers. 
In particular, if the operators S], S2, . . are not all of the same 
order those of highest order generate an invariant subgroup of G. If 
these orders represent more than two prime numbers, the operators 
of highest order together with those of the next lower order generate 
an invariant subgroup of G, etc. 
If the set of independent generators Si, S2, . . contains at least 
two non-commutating operators of the same order p, let Si, represent 
one of these operators which is commutative with the smallest number 
of the other operators of order p in the given set; and let S2, . . ., Sy 
represent all the operators of order p in the set 5i, ^2, . . . , S^, which are 
not commutative with Si. We proceed to determine the group generater 
by Si, S2, . . ., Sy. Since Si and S2 are non-commutative they generate 
a group of order pip, pi being a prime >p, which contains^! conjugate 
subgroups of order p. When y>2, Si and S3 will generate a similar 
group of order p2p. It may be assumed that pi^ p2' 
The group generated by Sj, S2 and S3 is of order pip2p' If pi =p2 
this group contains a single subgroup of order pi^, and each of its remain- 
ing operators is of order p since it contains no invariant operator besides 
the identity. If pj >p2, the group in question contains a cyclic subgroup 
of order p]p2 since the quotient group with respect to the invariant 
subgroup of order pj, cannot be cyclic. (See O. Holder, Gottingen 
Nachrichten, 1895, p. 298.) Hence the group generated by Si, S2, 
and S3 must always contain an abelian subgroup of index p whose 
order is prime to p, and each of its operators which is not contained 
in this subgroup must be of order p. 
When 7>3, we may consider separately the groups generated by 
Si, S2, S4 and by Si, S3, S4. The operators of order p3 in the group of 
order psp generated by and 5*4 must therefore be commutative with 
the operators of orders pj and p2 contained in the group generated by 
Si, S2 and 6*3. Hence it results that 6*1, ^2, ^3 and ^4 must generate a 
group having a single abehan subgroup of index p while all its remain- 
ing operators are of order p. As this mode of reasoning may clearly 
be continued until all the operators of the set 5], ^2, . . ., Sy have been 
exhausted, it results that these operators generate a group H having an 
abelian subgroup of index p whose order is prime to p, and that each of 
the other operators of H is of order p. In other words, H involves 
no invariant operator besides the identity, and the prime factors of 
the order of its subgroup of index p are all of the form 1 + kp. 
