MATHEMATICS: G. A. MILLER 
7 
Among the results of the present paper which are supposed to be new 
are the following: The number of the different operators in each of the 
possible sets of independent generators of a group whose order is a 
power of a prime number is the same, — that is, if the order of a group is 
a power of a prime number, the number of its independent generators is 
an invariant of the group. The 0-subgroup of every direct product is 
the direct product of the (^-subgroups of the factors of this direct product. 
In every group whose order is a power of a prime number the 0-sub- 
group includes the commutator subgroup of the group. By constructing 
the 0-subgroup of a group and of its successive </)-subgroup£ we can al- 
ways arrive at the identity. 
The (^-subgroup of a Sylow subgroup of the symmetric group of degree 
n is the commutator subgroup of this Sylow subgroup. In particular, 
the number of the different operators in a set of independent generators 
of the Sylow subgroup of order p''' of the S3anmetric group of degree p^, p 
being any prime number, is n. In particular, the subgroup of order 
pp+i which is contained in the symmetric group of degree p- has 
exactly 2 independent generators irrespective of the prime number 
represented by p. No (^-subgroup can contain a complete set of con- 
jugate subgroups, or a complete set of conjugate operators, involving 
more than one subgroup or more than one operator, when this complete 
set of conjugates is also a complete set of conjugates under the entire 
group. An important special case of this theorem is that every 0-sub- 
group whose order is not a power of a prime number is the direct product 
of its Sylow subgroups. This special case was noted by G. Frattini in 
the article to which reference has been given. 
With respect to abehan groups a set of X independent generators is 
commonly defined so that the group generated by every X - 1 of these 
generators has only the identity in common with the group generated 
by the remaining operator. For an abelian group whose order is a power 
of a prime number the number of the different operators in a possible 
set of independent generators is the same under both of the given defi- 
nitions of a set of independent generators. The fact that the number 
of independent generators of a group is not always an invariant of the 
group becomes evident if we observe that when we generate the sym- 
metric group of degree n by transpositions there will always be ^ - 1 
independent generators. On the other hand, this symmetric group can 
also be generated by a cyclic substitution of degree n - 1 and a trans- 
position involving the remaining letter. Complete proofs of these results 
are contained in a paper having the same heading, which has been offered 
for publication in the Transactions of the American Mathematical Society. 
