Finite Geometries

GEOMETRIES, FINITE, a class of geometries in each of which there is a finite number of (undefined) elements called points, falling into (undefined) subsets called lines, such that the mutual relations of lines and points (as well as various derived figures, such as planes, 3-spaces, etc.) are closely analogous to those of like objects in ordinary projective geometry. Follow ing Professors Veblen and Bussey, we define them in the following manner : A finite geometry consists of a set S of elements, called points for suggestiveness, which are subject to the following five condi tions or postulates : I. The set S consists of a finite number of points. It contains one or more subsets called lines, each of which contains at least three points.

II. If A and B are distinct points, there is one and only one line that contains both A and B. (We denote this line by AB.) III. If A, B, C are non-collinear points and if a line 1 contains Iii. If A, B, C are non-collinear points and if a line 1 contains a point D of the line AB and a point E of the line BC but does not contain A or B or C, then the line 1 contains a point F of the line CA.

If m is an integer less than k, not all the points considered li are in the same m-space.

Vic. When IV,, is satisfied, there exists in the set of points con sidered no (k+1)-space.

The geometry so defined is one of k dimensions.

In this system of postulates the terms point and line are left undefined. They are to be any objects of thought for which the postulates are true. An m-space, or a space of m dimensions, is then defined inductively as follows : A point is a 0-space and a line is a I-space. If ..., P,,,, 1 are points not all in the same (m— I)-space, then the set of all points each of which is collinear with and some point of the (m— I)-space P2, ..., P,,,) is the m-space P2, •••, A 2-space is called a plane.

It may be observed that the foregoing definitions and postu lates are all identical with corresponding definitions and proposi tions of the usual projective geometry, except for the one require ment here made that the number of points considered shall be finite. This is the justification for using the name geometry for the configurations here defined. Many of the propositions of projective geometry can be developed from postulates similar to the foregoing without saying whether the number of points is finite or infinite; and this is done in Veblen and Young's Pro jective Geometry. In fact this procedure is illuminating, so that the finite geometries justify themselves in the light which they throw on ordinary projective geometry. But they also have other uses, particularly in the development of the theory of permutation groups. Moreover, the theory in itself possesses such artistic ele gance as to commend it strongly to one who takes pleasure in such aspects of thought.

As an example of a finite geometry of two dimensions we have one in which the set S consists of the 13 letters A, B, C, ... M, while the lines consist of the 13 subsets of four elements each indicated by the columns of the following array: That this set of points, with the indicated subsets forming lines, satisfy the given postulates the reader may readily verify.

There are special finite geometries of two dimensions which cannot be extended to, or imbedded in, geometries of higher dimensions. These exceptional geometries we shall not consider. All our further statements will refer only to those geometries remaining after these exceptional cases are excluded.

Some Principal Theorems.

In any given finite geometry the number of points on one line is the same as the number on any other; and this number is always of the form r-{-pn where p is a prime and n is a positive integer. For every such p and n a unique k-dimensional finite (projective) geometry exists having I-}-pn points on a line; it is denoted by the symbol PG(k,pn). Many of its properties are common to it and the ordinary pro jective geometry of k-dimensional space. The number of points in a PG(k, pn) is . .+pkn If one omits from a PG (k, pn) any subspace of k— I dimensions, then there remains a set of pk ' points forming a Euclidean geom etry EG(k, pre) of k dimensions. It has many properties in com mon with the usual geometry of k dimensions and of Euclidean type. The finite geometries PG(k, pn) exhibit a property of duality in all respects analogous to the property of duality in ordinary projective geometry. Thus, in a space of three dimensions the planes may be interpreted as points if one at the same time re interprets the points as planes, while lines continue to be lines.

It is possible to introduce homogeneous co-ordinates into the geometry PG(k, pre) and thus to represent its points by means of k+I homogeneous co-ordinates. This is simplest in the case when n= I. For the PG(k, p) we employ for the number system the integers taken modulo p. Thus we have in this case the dis tinct numbers o, 2, p— I. In the general case of PG(k,pn) we use for the number system the marks of the Galois field GF(pn). In each case a point is denoted by the symbol µ,, ...., interpreted as a set of homogeneous co-ordinates, that is, interpreted so that µ, ... and (Pilo, vµ,, ..., Pilk) represent the same point if v is any mark different from zero; it is understood that one at least of the marks is,, .•., µk is different from zero. Then the points (xo, xl, .... xk,) whose co-ordinates satisfy one linear homogeneous equation = o constitute a space of k— I dimensions ; those satisfying two such equations (when these are independent) constitute a space of k— 2 dimensions ; and so on.

The geometry already exhibited as an example is the PG(2, 3); and its number system consists of integers taken modulo 3. It affords a convenient illustration of the results stated in the two foregoing paragraphs.

The totality of transformations of points by each of which a PG(k, pn) is left invariant, that is, transformations of points by which lines go into lines, are of particular importance. They are called collineations. This totality contains all the linear homo geneous transformations on k-}-i variables ..., into k+i variables (yo, yi, ..., yk), the set of variables in each case repre senting a point of the geometry. When n = i this totality contains all the collineations. When n> 1 it is necessary to adjoin the transformation yi=x1, i=o, 1, k, in order to generate all the collineations in PG(k, pi). Those transformations in this set, each of which leaves fixed a given (k— i)-space in PG(k, pn) transform among themselves the points of the corresponding EB(k, pn) according to the group of collineations of this Euclidean geometry.

Relations with Permutation

Groups.—The collineation groups give rise to some of the most interesting permutation groups, particularly those known as primitive groups. But the finite geometries have a more intimate relation than this to the gen eral theory of abstract finite groups. Thus an Abelian group H of order pk+i and type (1, 1, ..., i) affords a representation of the PG(k, p) by interpreting as points the subgroups of order p in H and as lines the subgroups of order This theorem, which is easily demonstrated, may be extended to the case of the geometries PG(k, pn) . Here we consider an Abelian group G of order and type (I, 1, ..., 1) and select from G a certain special set of subgroups of order p', interpreting each subgroup as a point, while a group of order pen generated by two of them is called a line. Then the collineation groups are interpretable by means of the holomorph and the group of isomorphisms of G. Thus the finite geometries may be treated as a chapter in group theory; and conversely all propositions in the finite geometries may be translated into propositions about Abelian groups.

See 0. Veblen and W. H. Bussey, Trans. Amer. Math. Soc., 7, 241-2S9 (1906) ; 0. Veblen and J. W. Young, Projective Geometry, vol. i. (1910) and vol. ii. (1918) (Boston, U.S.A.). (R. D. CA.)