MATHEMATICS: G. A. MILLER 
241 
^ See Dienger, Grundriss der Variationsrechnung, p. 27. 
« See Mayer, Leipzig, Ber. Ges. Wiss., 36, 99 (1884) and 48, 436 (1896); Bliss, Math. Ann., 
Leipzig, 58, 70 (1903); Erdmann, Zs. Math., Leipzig, 23, 364 (1878). 
' "On the reduction of a system of linear differential forms of any order," Annals Mathe- 
matics, 13, 149 (1912). In a paper entitled On the second variation, Jacobi's equation and 
Jacobi's theorem. Ibid., 15, 78 (1913), this reduction was used to unify the Weierstrass 
theory of the second variation and Jacobi's equation. 
GROUPS POSSESSING AT LEAST ONE SET OF INDEPENDENT 
GENERATORS COMPOSED OF AS MANY OPERATORS 
AS THERE ARE PRIME FACTORS IN THE 
ORDER OF THE GROUP 
By G. A. Miller 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF ILUNOIS 
Presented to the Academy, January 18, 1915 
If ^i, 52, . . Sy^ represent a set of operators of the group G such 
that these X operators generate G but that no X — 1 of them generate G. 
then these X operators are called a set of independent generators of G. 
Every subset of a set of independent generators of any group whatever 
generates a subgroup whose order contains at least as many prime 
factors as the number of operators in this subset. In particular the 
number of operators in a set of independent generators never exceeds 
the number of prime factors in the order of the group except in the 
trivial case when the group is the identity, unity not being regarded as 
a prime factor. 
If G represents the abelian group of order, and of type (1,1,1,...), 
it is evident that each of its possible sets of independent generators 
involves as many operators as there are prime factors in the order of G. 
In what follows we propose to determine some properties possessed by 
all those groups which have at least one set of independent generators 
composed of as many operators as there are prime factors in the order 
of the group. The symbol G will hereafter in the present article be 
used to represent any one of these groups, and we shall assume in what 
follows that 5], ^2, • • J always represent a set of independent genera- 
tors of G such that X is equal to the number of the prime factors of the 
order g oi G. 
Each of the operators Si, S2, . . ., must be of prime order since 
each subset of these independent generators generates a subgroup 
whose order has exactly as many prime factors as the number of oper- 
ators in this subset. A necessary and sufficient condition that the 
groups generated by two such subsets have only the identity in common 
