The simplest four-dimensional manifoldnesses in three-dimensional space are that of all straight lines and that of all spheres. For this reason those contact transformations between two three-dimensional spaces or which change a three-dimensional space into itself in such a manner that straight lines are changed into spheres, are the first to attract attention and have so far been the most fruitful. Lie constructed such a transformation in his memoir on complexes in the fifth volume of the Mathematische Annalen which has led him to a generalized form of the theorem of Malus.

*

Lately this manner of changing straight lines into spheres by contact transformations has been found not to be unique; in fact infinite groups of infinite numbers of such line-sphere contact transformations have been constructed.

The above observations increase the demand for the resolution of the problem of determining all continuous groups in four variables. But such contact transformations need not necessarily be contact transformations of a three-dimensional point space into itself; for example, if the four variables be interpreted as line-coordinates or spherecoordinates, the corresponding invariant Pfaffians by no means provide that the conditions for contact transformations of the three-dimensional space into itself be satisfied. It is precisely because of such a confusion that we find these notions used loosely in a recent memoirt on the employment of infinitesimal transformations in optics.

"Lichtstrahlen, die in Pseudonormalensystem bilden, gehen bei jeder Reflexion und Refraction in ein Pseudonormalensystem über. Sind bei einer solchen Refraction die beiden in Betracht kommenden Pseudokugeln (d. h. Wellenflächen) wesentlich verschieden, so bezieht sich jedes

Pseudonormalensystem auf die Pseudokugel des betreffenden
Raumes," Leipziger Berichte, 1896, loc. cit., p. 133.

+ Hausdorff, "Infinitesimale Abbildungen der Optik," Leipziger Berichte, 1896, pp. 79-130.

XI.

On a Class of Groups of Finite Order. By Professor W. BURNSIDE.

[Received 30 September 1899.]

AMONG the groups of finite order that earliest present themselves, from some points of view, to the student are the groups of rotations of the regular solids. An admirable account of these from the purely geometrical stand-point is given in the first chapter of Klein's Vorlesungen über das Icosaeder. Of the six types included in this set of groups there are three which, though quite unlike in other respects, have a distinctive property in common. These are (i) the dihedral group of order 2n (n odd), (ii) the tetrahedral group of order 12, and (iii) the icosahedral group of order 60. They are defined abstractly by the relations :

The order of each of these groups is even, while the only operations of even order which they contain are operations of order two. While they have this property in common they are otherwise of very distinct types.

The first has an Abelian (cyclical) self-conjugate subgroup, order n, which consists of the totality of its operations of odd order. The second contains a self-conjugate subgroup of order four, this being the highest power of two which is a factor of the order of the group. The third is a simple group containing five subgroups of order twelve, each of which has a self-conjugate subgroup of order four. It can be represented as a triply-transitive substitution group of degree five.

I propose here to determine the groups of even order, which contain no operations of even order other than operations of order two. The determination is exhaustive; and it will be seen that the groups in question arrange themselves in three quite different sets of types of which the groups (i), (ii) and (ii), defined above, are representative.

1. Let G be a group of even order N, which contains no operations of even order

other than those of order two. To deal first with the simplest case that presents itself*, let

N = 2m,

where m is odd. Since no operation of order two is permutable with any operation of odd order, G must contain m operations of order two which form a single conjugate set. Let these be

If AA, were an operation of order two, 1, Ar, A,, and A,A,, would constitute a subgroup of G of order four. No such subgroup can exist, and therefore A, A, is an operation of odd order. The m operations

which are necessarily distinct, are therefore the m operations of odd order contained in G. These m operations may similarly be expressed in the form

A, transforms every operation of G, of odd order, into its inverse. Hence

and this shews that every pair of operations of G, of odd order, are permutable. Hence the m operations of G of odd order, including identity, constitute an Abelian group, and this is a self-conjugate subgroup of G. Conversely, if H is any Abelian group of odd order m, generated by the independent operations S, T, and if A is an operation of order two such that

ASAS, ATA=T1,

then A and H generate a group G of order 2m, whose only operations of even order are those of order two.

When r is given, s can always be taken in just one way so that A.A, is any given operation of G of odd order. Hence every operation of G of odd order can be represented in the form A,A, in just m distinct ways. This property will be useful in the sequel.

The groups thus arrived at are obviously analogous to the group (i) above.

where m is odd and n is greater than one. The operations of order two contained in G form one or more conjugate sets. Suppose first that they form more than one such set; and let

* This first case is considered in my Theory of Groups of Finite Order, pp. 143 and 230.

be two distinct conjugate sets of operations of order two. The operation AB must either be of order two or of odd order. If it were of odd order, μ, the subgroup generated by A and B would be a dihedral subgroup of order 2μ; and in this subgroup A and B would be conjugate operations. operations. Since A and B belong to distinct conjugate sets in G, this is impossible. Hence AB is of order two, or in other words A and B are permutable. Every operation of one of the two conjugate sets is therefore permutable with every operation of the other. The two conjugate sets therefore generate two self-conjugate subgroups (not necessarily distinct) such that every operation of the one is permutable with every operation of the other. The order of each of these is divisible by two, and therefore the order of each must be a power of two; as otherwise G would contain operations of order 2r (r odd). The two together will generate a self-conjugate subgroup H' of order 2". If n' is less than n, there must be one or more conjugate sets of operations of order two not contained in H'. Let

c, c', ...,

be such a set. As before every operation of this set must be permutable with every operation of H'. Hence finally G must contain a self-conjugate subgroup H of order 2". No operation of G is permutable with any operation of H except the operations of H itself; and G is therefore a subgroup of the holomorph* of H. It follows that G can be represented as a transitive group of degree 2". Moreover, since G contains no operations of even order except those of order two, the substitutions of this transitive group must displace either all the symbols or all the symbols except Hence m must be a factor of 2′′-1; and G contains 2" subgroups of order m which have no common operations except identity. With the case at present under consideration may be combined that in which G has a self-conjugate subgroup of order 2", the 2′′ – 1 operations of order two belonging to which form a single conjugate set. In this case m must be equal to 2′′ – 1.

one.

We thus arrive at a second set of groups with the required property of order 2am, where m is equal to or is a factor of 2"-1. They have a self-conjugate subgroup of order 2", and 2" conjugate subgroups of order m; the latter having no common operations except identity. These are clearly analogous to group (ii) above.

3. Lastly there remains to be considered the case in which the operations of G of order two form a single conjugate set, while G contains more than one subgroup

of order 2n.

If H and H are two subgroups of G of order 2", and if I is the subgroup common to H and H', then since H and H' are Abelian (their operations being all of order two) every operation of I is permutable with every operation of the group generated by H and H'. This group must have operations of odd order, since it contains more than one subgroup of order 2". Hence I must consist of the identical operation only; or in other words, no two subgroups of order 2" have common operations other Theory of Groups, p. 228.

*

than identity.

It follows from an extension of Sylow's theorem that the number of subgroups of order 2′′ contained in G must be of the form 2k + 1.

If K is the greatest subgroup of G which contains a subgroup H, of order 2", self-conjugately; then K must be a subgroup of the nature of those considered in the preceding section, and its order must be 2"μ, where μ is equal to or is a factor of 2n-1. Also no two operations of H can be conjugate in G unless they are conjugate in K*. The 2′′-1 operations of order two in K therefore form a single conjugate set; and hence must be equal to 2" - 1. The order of G is therefore given by

N = (2nk + 1) 2′′ (2′′ — 1).

That G must be a simple group is almost obvious. A self-conjugate subgroup of even order must contain all the 2k+1 subgroups of order 2", since the operations of order two form a single set. In such a subgroup the operations of order two must form a single set, and therefore a subgroup of order 2" must be contained self-conjugately in one of order 2" (2" - 1). Hence a self-conjugate subgroup of even order necessarily coincides with G. If on the other hand G had a self-conjugate subgroup I of odd order r, I would by the first section be Abelian and every operation of G of order two would transform every operation of I into its inverse. This is impossible; for if A and B were two permutable operations of order two in G which satisfy the condition, then AB is an operation of order two which is permutable with every operation of I, contrary to supposition. Hence G must be simple.

If A and B are any two non-permutable operations of order two in G, AB must be an operation of odd order μ, and A and B generate a dihedral group of order 2μ. Hence G contains subgroups of the type considered in the first section. Let 2m, be the greatest possible order of a subgroup of this type contained in G; and let I, be a subgroup of G of order 2m1, and J, the Abelian subgroup of order m, contained in I. Every subgroup K of J is contained self-conjugately in I1; and, for the reason just given in proving that G is simple, no two permutable operations of order two can transform K into itself. Hence I must be the greatest subgroup that contains K self-conjugately; as otherwise 2m, would not be the greatest possible order for the subgroups of this type contained in G.

Let pa be the highest power of a prime p which divides m,; and let K be a subgroup of J1 of order pa. If pa is not the highest power of p which divides N, then K would be contained self-conjugately† in some subgroup of G of order pa+1. This has been proved impossible. Hence m, and N/m, are relatively prime.

1

Again no two subgroups conjugate to J, can contain a common operation other than identity; for if they did I would not be the greatest subgroup of its type contained in G.

If I, and the subgroups conjugate to it do not exhaust all subgroups of G of order 2μ (μ odd), let I, of order 2m, (m, odd) be chosen among the remaining subgroups of G of