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MATHEMATICS: G. A. MILLER 
of order prime to p, while all its other operators are of order p; (3) Group 
which is the direct product of such groups as are defined in (1) and (2). 
When no operator in the given set of independent generators of G 
has an order which exceeds p, it results that the group generated by all 
the operators of order p in this set is invariant under G. When this 
set contains non-commutative operators of order p all of these opera- 
tors which are non- commutative with some one of them generate an 
invariant subgroup, composed of operators which are commutative 
v/ith every operator in the group generated by all the operators of the 
set except those used to generate this invariant subgroup. As an im- 
portant but very special result we may mention the theorem that every 
group which contains a set of independent generators composed of as many 
operators as there are prime factors in the order of the group is solvable. 
Since each operator in the abelian invariant subgroup of H is commu- 
tative with each of the operators of the set S\,S2, . . ., Sx, except those 
which are contained in H, it is easy to see that every Sylow subgroup of 
G is abelian and of type (1, 1, 1, . . .). In fact, if all the operators of 
the same order in the set Si, S2, . . S\ are commutative the various 
subsets composed of the operators of the same order in this set would 
generate such Sylow subgroups of G. If not all the operators of the 
same order in Si, S2, . . ., Sx are commutative then these non-commu- 
tative operators may be divided into subsets such that each subset 
generates a group having the properties which have been proved to 
belong to H. Hence the theorem: // a group contains at least one set 
of independent generators composed of as many operators as there are 
prime factors in the order of the group then all of its Sylow subgroups are 
abelian and of type (1, 1, I,...)- As a special case of this theorem we 
have the known and evident result that when the order of such a group 
is a power of a prime the group must be abelian and of type (1, 1, 1,. . .)• 
Among the important categories of groups which have the property 
that it is possible to hnd a set of independent generators composed 
of as many operators as there are prime factors in the order of the group 
is the important category of solvable groups composed of all the groups 
whose order is not divisible by the square of a prime number. If the 
order of such a group involves more than one prime factor, it is always 
possible to select a set of independent generators composed of an arbi- 
trary number of operators from 2 to the number of prime factors in the 
order of the group. This fact can be deduced directly from the theorem 
due to Holder to which we referred above. This important category 
of solvable groups is therefore included in the larger category of such 
groups considered above. 
