MATHEMATICS: G. A. MILLER 
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All the operators of G besides the identity must be of order 2 or of order 4 
and G must involve operators of each of these two orders. Hence the order 
of G is of the form 2^. When G is abelian it is of type (2, 1, 1, . . . ) and 
it will therefore be assumed in what follows that G is non-abehan. The octic 
group and the quaternion group constitute well known illustrations of such a 
group and have the smallest possible order. 
When the operators of order 2 contained in G together with the identity 
constitute a subgroup this subgroup is the central of G and hence G belongs to 
the system of groups called Hamiltonian by R. Dedekind.^ In this case it is 
known that G is the direct product of the quaternion group and an abelian 
group of order 2" and of type (1,1,1, . . . ). Hence it will be assumed in 
what follows that G involves non-commutative operators of order 2. 
Every operator of order 4 contained in G is transformed either into itself 
or into its inverse by every operator of G and an operator of order 2 contained 
in G has at most two conjugates under the group.^ Let H], H2 represent sub- 
groups composed respectively of all the operators of G which are commuta- 
tive with two non-commutative operators of order 2 ^1, ^2. The cross-cut Ki 
of El and H2 is of index 4 under G and includes the central of G. A set of in- 
dependent generators of G can be so selected as to include Si, S2 and operators 
from K\. 
Exactly one-half of the operators of G which are not also in Ki, are of order 
2 since the quotient group G/Ki is abehan. If Ki involves non-commutative 
operators of order 2 two such operators sz, may be selected from Ki in ex- 
actly the same way as Si and S2 were selected from G. The remaining opera- 
tors of a set of independent generators including ^i, ^2, -^s, -^4 may be selected 
from an invariant subgroup of index 4 under Ki and of index 16 under G all 
of whose operators are commutative with each of the four operators already 
chosen. 
As G is supposed to be of finite order we arrive by this process at a sub- 
group in which all the operators of order 2 are commutative. Hence 
belongs to one of the following three well known categories of groups. Abe- 
lian and of type (1, 1, 1, . . . ), abelian and of type (2, 1, 1, . . . ), or 
Hamiltonian of order 2". The commutator subgroup of G is of order 2. 
In each case, G may be constructed by starting with K^, forming the direct 
product of and an operator h of order 2, and then extending this direct 
product by means of an operator t2 of order 2 which is commutative with each 
of the operators of Z"^ and transforms ti into itself multipHed by the commuta- 
tor of order 2 contained in G. When K^^ is Hamiltonian or abehan and of 
type (2, 1, 1, . . . ) this commutator is determined by K^. In the other 
possible case it may be selected arbitrarily from the operators of order 2 found 
in K^. 
When m> 1, we use the group K^_i just constructed in exactly the same way 
as Km was used in the preceding paragraph. The commutator of order 2 is 
completely determined for each of the categories by K,^_i, m>\. When m>2 
