Videns piscator a militibus se comprehendi putavit occidi. Ingressus Appollonius jussit eum adduci ad se et ait: "hic est paranymphus meus, qui mihi post naufragium opem dedit et ad civitatem venire ostendit." Et dicit ei: "ego sum Tyrius Appollonius." Et jussit sibi dari CC sistercias argenti, servos et ancillas, et fecit eum comitem suum, quamdiu vixit. Elamitus vero, qui ei de Antiocho nunciavit, procidens ad pedes Appollonii .... et ait: "domine, memor esto Elamiti servi tui!" Appollonius apprehensa manu eius erexit eum fecitque eum divitem et ordinavit comitem. Hiis expletis genuit Appollonius filium de conjuge sua, quem in loco avi sui Altistratis constituit regem. Vixit vero Appollonius cum conjuge sua annos LXXIV et tenuit regnum Antioche et Tyri et Tyrenensium quiete ac feliciter. Casus suos ipse descripsit, ipse duo volumina perfecit, unum in templo Ephesiorum, alterum in sua bibliotheca collocavit. Et defunctus est et perrexit ad vitam eternam, ad quam vitam nos perducat, qui sine fine vivit et regnat Amen.

ON THE QUATERNION GROUP.

BY G. A. MILLER, PH.D.

(Read October 7, 1898.)

Although the quaternion group (Q) has received some attention,1 yet many of the properties of this important group remain to be investigated. It is the object of this paper to enter upon the study of some of these group properties after stating the known principles which underlie the investigations that follow. We shall also determine the different ways in which may be represented as a substitution group.

It is well known that every group of a finite order may be represented as a regular substitution group and that any two regular substitution groups which are simply isomorphic are also conjugate.

A complete list of the regular substitution groups of order g must therefore include every possible group of this order and no group can occur twice in such a list. In following Prof. Cayley's

1 Dedekind, Mathematische Annalen, 1897, Vol. xlviii, pp. 549–552.

notation we represent Q as a regular substitution group in the following manner :

1

THE DIFFERENT WAYS IN WHICH MAY BE Represented AS A SUBSTITUTION GROUP.

We observe, in the first place, that Q cannot be represented as a non-regular transitive substitution group. If such a representation were possible Q would have to contain some subgroup of a prime order that is not self-conjugate. As it contains only one subgroup of order 2 this must clearly be self-conjugate. Hence we observe that there is only one transitive substitution group that is simply isomorphic to Q.

It is known that the number of the intransitive substitution groups that are simply isomorphic to a given group is an increasing function of the degree, which becomes infinite when the degree becomes infinite. We proceed to determine the nature of this function in the present case. Since every group whose order is the square of a prime number is Abelian, a substitution group which is simply isomorphic to must contain at least one transitive constituent of order 8 and its degree must be 2 n, n being a positive integer greater than 3.

We have seen that Q contains only one subgroup of order 2. With respect to this it is isomorphic to the four-group, since this subgroup contains the square of each one of its operators. As a subgroup whose order is one-half of the order of the entire group must always be self-conjugate, Q contains three self-conjugate subgroups of order 4. Since none of these three subgroups is characteristic they must be transformed into each other by the largest

1 Cayley, Quarterly Journal of Mathematics, 1891, Vol. xxv, p. 144. 2 Cf. Dyck, Mathematische Annalen, 1883, Vol. xxii, p. 90. It may be remarked that the statement on p. 101 of this article that a group which can be represented only in the regular form contains only self-conjugate subgroups is not quite correct, as may also be inferred from other parts of the same article. 3 Frobenius, Berliner Sitzungsberichte, 1895, p. 183.

group that contains Q as a self-conjugate subgroup. Hence we need to consider only one of these three subgroups in connection with the study of the intransitive substitution groups that are simply isomorphic to Q.

We may now state the problem of finding all the substitution groups that are simply isomorphic to Q in the following manner. Such a group contains a transitive constituents of order 8, where a is an integer greater than o. Its other constituents form a group whose order is either 4 or 2. If this order is four these constituents must form the four-group. If it is two these can form only one group for a given set of values of n and a. Hence we observe that the number of quaternion substitution groups of degree 2 n, n> 3, which contain no constituent group of order 4 is a,, where a is the largest integral value of x that satisfies the relation:

n

4

To find the number of these groups that contain a constituent of order 4 we may first find the number of those that contain only one transitive constituent of order 8, then the number of those that contain two such constituents, etc. The sum of these numbers is the number required. Each of these numbers may be directly found by means of the following formula,' in which N is the number of all the possible substitution groups of order 4 and degree 2n, m is any positive integer, and a, is the largest value of y that satisfies the relation

66 n =

6 m + 2, N3m (m + 1) (m + 2) + 1 + α,

66

n = 6 m + 4, N= (m + 1) (3 m2 + 9 m + 4) + α1

1 Miller, Philosophical Magazine, 1896, Vol. xli, p. 437.

If we add a, to the sum of the numbers obtained by means of these formulas we obtain the total number of the substitution groups of degree 2n that are simply isomorphic to Q. Among these substitution groups the given regular group is especially convenient for the study of the properties of Q.

In what follows we shall, therefore, suppose written in this way unless the contrary is explicitly stated.

It is known that all the substitutions that involve no more than g letters and are commutative to every substitution of a regular group involving the same g letters form a group which is conjugate to the regular group. This conjugate of the given regular group contains the following substitutions:

One of the 192 substitutions in these 8 letters that transform one of these two regular groups into the other is the transposition dh.

THE GROUP OF ISOMORPHISMS OF Q.

The largest group in these eight letters that transforms one of the two given regular groups into itself must be transitive, since it includes a regular group. Its subgroup which includes all its substitutions that do not involve a given letter is the group of isomorphisms of Q. We proceed to prove that this is simply isomorphic to the symmetric group of order 24. To prove this we observe that an operator of order 4 may be made to correspond to any other operator of this order in a simple isomorphism of Q to itself. Hence the first correspondence can be effected in 6 ways and the second can evidently be effected in 4 ways, so that the group of isomorphisms must be of order 24.

This group of isomorphisms may be represented as a transitive substitution group of degree 6, since there are 6 operators of order

4 that can be made to correspond and these generate Q. As this substitution group cannot contain a substitution whose degree is less than 4 and the transitive groups of degree 6 and order 24 that ́ have this property are simply isomorphic to the symmetric group of this order it follows directly that the group of isomorphisms of Qis the symmetric group of order 24 and that the group of cogredient iscmorphisms is its self-conjugate subgroup of order 4.

There are two transitive groups of degree 6 that are simply isomorphic to the symmetric group of order 24. In one of these the subgroup which contains all the substitutions that do not include a given element is the cyclical group of order 4 while in the other it is the four-group. It remains to determine which of these two groups is the substitution group of isomorphisms of Q. This may be easily done by making Q simply isomorphic to itself in the following manner:

The substitution which corresponds to this isomorphism is given by the second columns of letters; hence it is bdfh and the substitution group of isomorphisms of Q is the one which Prof. Cayley represents by (± abcdef)24.1

As

It is known that is simply isomorphic to the eight unities (1, —1, i, —i, j, —j, k, —k) of the quaternion number system. Q can be made simply isomorphic to itself in 24 different ways the simple isomorphism of Q to these unities or of these unities to themselves may also be written in 24 ways. The following is one of these ways:

It may be very easily verified that the following relations are

1 Quarterly Journal of Mathematics, 1891, Vol. xxv, p. 80.