Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

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HAMILTON GROUPS

117

The group generated by any one such quaternion group and
the operators of order 2 contained in H must coincide with H.
In fact, if an operator s of H were not contained in this group,
this operator and this group would generate a group which
would involve more operators of order 2. As this is contrary
to the hypothesis, we have established the following theorem:
Every possible Hamilton group is the direct product of a quaternion
group, an abelian group of order 2" and of type (1, 1, 1,.. ),
and an abelian group of odd order.*
A group which is a direct product of two groups is sometimes called a divisible group. If it is not a direct product
it is said to be indivisible. Hence the quaternion group is the
only indivisible Hamiltonian group. The only indivisible abelian
groups are the cyclic groups whose orders are powers of prime
numbers. If an abelian group is written as the product of indivisible groups, the orders of these groups constitute the largest
possible set of invariants of the abelian group.
EXERCISES
1. The commutator quotient group of a Hamilton group of order 2"
is abelian and of type (1, 1, 1... ).
2. The number of the possible Hamilton groups of order 2mk, k being
any odd number, is equal to the number of the abelian groups of order k.
3. Every Hamilton group has the four-group as a group of inner isomorphisms.
4. Two and only two of the operators of a Hamilton group are characteristic.
5. Let g=pl p2a2.. pxa, where pi, P2,., pX are distinct primes.
Necessary and sufficient conditions that all existing groups of order g
shall be abelian are: (1) each aj< 2; (2) no pjai-1 is divisible by one
of the primes pi, P2,.., Px. Cf. L. E. Dickson, Transactions of the
American Mathematical Society, vol. 6 (1905), p. 201.
* This theorem, together with various other theorems relating to Hamilton
groups, was proved by G. A. Miller, Bulletin of the American Mathematical
Society, vol. 4 (1898), p. 510. Some of these theorems were proved several years
later by E. Wendt, Mathematische Annalen, vol. 59 (1904), p. 187. In the following volume of this journal Wendt corrected this oversight.