274 
MATHEMATICS: G. A. MILLER 
we proceed in the same manner with i^w_2, etc. It may be noted that in each 
of the groups belonging to one of the three categories thus constructed more 
than one-half of the operators are of order 2, in those belonging to the second 
category the number of operators of order 2 is one less than one-half of the 
order of while in those belonging to the third category the number of op- 
erators of order 2 is obtained by subtracting from one-half the order of G one 
plus one-fourth the order of K^. 
Some of these results constitute a proof of the following theorem: // only 
two of the operators of a group G are the squares of operators contained in G then 
the non-invariant operators of G have only two conjugates, each cyclic subgroup 
of order 4 is invariant, and G belongs to one of three categories of groups of order 
2 which can be separately generated by a set of operators such that each of these 
operators is commutative with each of the others except at most one of them. 
When m is sufficiently large there is one and only one group belonging to 
each of these three categories and having a give number y of pairs of non- 
commutative operators of order 2 in its set of independent generators when 
this set is obtained in the manner described above. The smallest values of 
m for these categories are 27 + 1, 27 -f 2, and 27 -f- 3 respectively. When 
m has a larger value G must be the direct product of an abelian group of type 
(1, 1, 1, . . . ) and of the minimal group having 7 such pairs of generators 
and contained in the category to which G belongs. 
By means of these facts it is very easy to determine the number of the groups 
of a given order 2"* which belong to each of these three categories. This 
number is the largest integer which does not exceed """2"' ~2~ 
for the three categories respectively. In particular, the number of the dis- 
tinct groups of order 128 belonging to each of these categories is 3, 2, 2 re- 
spectively, it being assumed that each of the groups in question contains at 
least two non-commutative operators of order 2. 
In each one of these groups every two non-commutative operators of order 
2 generate the octic group and every two non-commutative operators of order 
4 generate the quaternion group. Moreover, every non-abelian subgroup is 
invariant. In two of the categories the central is composed of operators of 
order 2 in addition to the identity, while the central of the remaining category 
is of type (2, 1, 1, . . . ). Every one of these groups is generated by its 
operators of order 2. From the standpoint of definition and structure these 
categories rank among the simplest known infinite systems of non-abelian 
groups 
1 Dedekind, R., Math. Ann., Leipzig, 48, 1897, (548-561). 
2 MiUer, G. A., Trans. Amer. Math. Soc, New York, 8, 1907, (1-13). 
