MATHEMATICS: G. A. MILLER 
243 
Since was so selected as to be commutative with the smallest 
possible number of operators of order p in the set 6*1, S2, . . S^^ it 
results that each of the operators 8^,82, . . Sy is commutative with 
every other operator of order p in this set. Each of the operators of 
H whose order exceeds p must be commutative with every one of the 
independent generators Syj^i, . . S^. In fact, if the order of such an 
independent generator exceeds p, it must be transformed into a power 
of itself by all the operators of H, since it is transformed into a power 
of itself by each of the operators Si, S2, . . Sy, which generate H. 
It results therefore from the theorem of Holder, to which we referred 
above, that the independent generator in question is commutative 
with every operator of H whose order exceeds p. 
On the other hand, every operator of the set Si, S2, . . .,S^ whose order 
is less than p, must transform each of the operators Si, . . Sy into a 
power of itself. This must be the first power for all of these operators. 
In fact, if Sa, l^a^y, were not transformed into its first power by such 
an operator S^ then the group generated by S^^ and some other operator 
of the set Si, . . ., Sy would be transformed into itself by S13, This is 
clearly impossible in view of the Holder theorem to which we referred 
above. Hence it results that every operator of H is commutative 
with every operator of the group generated by those operators of the 
set Sy^i, . . ., S^ whose orders do not exceed p. 
Some of these results may be expressed in the form of a theorem as 
follows : // a set of independent generators, which involves as many operators 
as there are prime factors in the order of G, includes at least two non- 
commutative operators of the same order p then all the operators of order p 
in this set, which are non-commutative with some one of them, generate a 
group H involving an ahelian invariant subgroup of index p under H, 
and every operator of this invariant subgroup is commutative with every 
operator of the group generated by all the operators of the given set of in- 
dependent generators of G, which are not contained in H. It results also 
directly from the preceding developments that if H does not include 
all the operators of order p in the given set of independent generators 
of G, then these remaining operators of order p may be combined into 
subsets such that all the operators of each subset generate either an 
abelian group of order p^ and of type (1, 1, 1, ..),ora group having the 
properties which we proved to apply to H. Hence the theorem. 
// a group G contains a set of independent generators composed of as 
m.any operators as there are prime factors in the order of G then all the oper- 
ators of the same order p in this set generate one of the following three 
groups: (1) Abelian groups of order p^ and of type (1, 1, 1,...); (2) 
Non-abelian group having an abelian invariant subgroup of index p but 
