Chapter II: Front Matter (2)

In the classification of the groups, projective or non-projective of
two or more variables, the distinction between primitive and
imprimitive groups immediately presents itself. For groups of the
plane the following question arises. Is there or is there not a
singly-infinite family of curves [f](x, y) = C, where C is an
arbitrary constant such that every operation of the group interchanges
the curves of the family among themselves? In accordance with the
previously given definition of imprimitivity, the group is called
imprimitive or primitive according as such a set exists or not. In
space of three dimensions there are two possibilities; namely, there
may either be a singly infinite system of surfaces F(x, y, z) = C,
which are interchanged among themselves by the operations of the
group; or there may be a doubly-infinite system of curves G(x, y, z) =
a, H(x, y, z) = b, which are so interchanged.

In regard to primitive groups Lie has shown that any primitive group
of the plane can, by a suitably chosen transformation, be transformed
into one of three definite types of projective groups; and that any
primitive group of space of three dimensions can be transformed into
one of eight definite types, which, however, cannot all be represented
as projective groups in three dimensions.

The results which have been arrived at for imprimitive groups in two
and three variables do not admit of any such simple statement.

Contact transformations.

We shall now explain the conception of contact-transformations and
groups of contact-transformations. This conception, like that of
continuous groups, owes its origin to Lie.

From a purely analytical point of view a contact-transformation may be
defined as a point-transformation in 2n + 1 variables, z, x1, x2, ...,
x_n, p1, p2, ..., p_n which leaves unaltered the equation dz - p1dx1 -
p2dx2 - ... - p_ndx_n = 0. Such a definition as this, however, gives
no direct clue to the geometrical properties of the transformation,
nor does it explain the name given.

In dealing with contact-transformations we shall restrict ourselves to
space of two or of three dimensions; and it will be necessary to begin
with some purely geometrical considerations. An infinitesimal
surface-element in space of three dimensions is completely specified,
apart from its size, by its position and orientation. If x, y, z are
the co-ordinates of some one point of the element, and if p, q, -1
give the ratios of the direction-cosines of its normal, x, y, z, p, q
are five quantities which completely specify the element. There are,
therefore, [oo]^5 surface elements in three-dimensional space. The
surface-elements of a surface form a system of [oo]^2 elements, for
there are [oo]^2 points on the surface, and at each a definite
surface-element. The surface-elements of a curve form, again, a system
of [oo]^2 elements, for there are [oo]^1 points on the curve, and at
each [oo]^1 surface-elements containing the tangent to the curve at
the point. Similarly the surface-elements which contain a given point
clearly form a system of [oo]^2 elements. Now each of these systems of
[oo]^2 surface-elements has the property that if (x, y, z, p, q) and
(x + dx, y + dy, z + dz, p + dp, q + dq) are consecutive elements from
any one of them, then dz - pdx - qdy = 0. In fact, for a system of the
first kind dx, dy, dz are proportional to the direction-cosines of a
tangent line at a point of the surface, and p, q, -1 are proportional
to the direction-cosines of the normal. For a system of the second
kind dx, dy, dz are proportional to the direction-cosines of a tangent
to the curve, and p, q, -1 give the direction-cosines of the normal to
a plane touching the curve; and for a system of the third kind dx, dy,
dz are zero. Now the most general way in which a system of [oo]^2
surface-elements can be given is by three independent equations
between x, y, z, p and q. If these equations do not contain p, q, they
determine one or more (a finite number in any case) points in space,
and the system of surface-elements consists of the elements containing
these points; i.e. it consists of one or more systems of the third
kind.

If the equations are such that two distinct equations independent of p
and q can be derived from them, the points of the system of
surface-elements lie on a curve. For such a system the equation dz -
pdx - qdy = 0 will hold for each two consecutive elements only when
the plane of each element touches the curve at its own point.

If the equations are such that only one equation independent of p and
q can be derived from them, the points of the system of
surface-elements lie on a surface. Again, for such a system the
equation dz - pdx - qdy = 0 will hold for each two consecutive
elements only when each element touches the surface at its own point.
Hence, when all possible systems of [oo]^2 surface-elements in space
are considered, the equation dz - pdx - qdy = 0 is characteristic of
the three special types in which the elements belong, in the sense
explained above, to a point or a curve or a surface.

Let us consider now the geometrical bearing of any transformation x' =
[f]1(x, y, z, p, q), ..., q' = [f]5(x, y, z, p, q), of the five
variables. It will interchange the surface-elements of space among
themselves, and will change any system of [oo]^2 elements into another
system of [oo]^2 elements. A special system, i.e. a system which
belongs to a point, curve or surface, will not, however, in general be
changed into another special system. The necessary and sufficient
condition that a special system should always be changed into a
special system is that the equation dz' - p'dx' - q'dy' = 0 should be
a consequence of the equation dz - pdx - qdy = 0; or, in other words,
that this latter equation should be invariant for the transformation.

When this condition is satisfied the transformation is such as to
change the surface-elements of a surface in general into
surface-elements of a surface, though in particular cases they may
become the surface-elements of a curve or point; and similar
statements may be made with respect to a curve or point. The
transformation is therefore a veritable geometrical transformation in
space of three dimensions. Moreover, two special systems of
surface-elements which have an element in common are transformed into
two new special systems with an element in common. Hence two curves or
surfaces which touch each other are transformed into two new curves or
surfaces which touch each other. It is this property which leads to
the transformations in question being called contact-transformations.
It will be noticed that an ordinary point-transformation is always a
contact-transformation, but that a contact-transformation (in space of
n dimensions) is not in general a point-transformation (in space of n
dimensions), though it may always be regarded as a
point-transformation in space of 2n + 1 dimensions. In the analogous
theory for space of two dimensions a line-element, defined by (x, y,
p), where 1 : p gives the direction-cosines of the line, takes the
place of the surface-element; and a transformation of x, y and p which
leaves the equation dy - pdx = 0 unchanged transforms the [oo]^1
line-elements, which belong to a curve, into [oo]^1 line-elements which
again belong to a curve; while two curves which touch are transformed
into two other curves which touch.

One of the simplest instances of a contact-transformation that can be
given is the transformation by reciprocal polars. By this
transformation a point P and a plane p passing through it are changed
into a plane p' and a point P' upon it; i.e. the surface-element
defined by P, p is changed into a definite surface-element defined by
P', p'. The totality of surface-elements which belong to a
(non-developable) surface is known from geometrical considerations to
be changed into the totality which belongs to another
(non-developable) surface. On the other hand, the totality of the
surface-elements which belong to a curve is changed into another set
which belong to a developable. The analytical formulae for this
transformation, when the reciprocation is effected with respect to the
paraboloid x^2 + y^2 - 2z = 0, are x' = p, y' = q, z' = px + qy - z,
p' = x, q' = y. That this is, in fact, a contact-transformation is
verified directly by noticing that dz' - p'dx' - q'dy' = -d(z - px -
qy) - xdp - ydq = -(dz - pdx - qdy). A second simple example is that
in which every surface-element is displaced, without change of
orientation, normal to itself through a constant distance t. The
analytical equations in this case are easily found in the form

pt qt
x' = x + ---------------------, y' = y + --------------------,
[root](1 + p^2 + q^2) [root](1 + p^2 + q^2)

t
z' = z - ---------------------,
[root](1 + p^2 + q^2)

p' = q, q' = q.

That this is a contact-transformation is seen geometrically by
noticing that it changes a surface into a parallel surface. Every
point is changed by it into a sphere of radius t, and when t is
regarded as a parameter the equations define a cyclical group of
contact-transformations.

The formal theory of continuous groups of contact-transformations is,
of course, in no way distinct from the formal theory of continuous
groups in general. On what may be called the geometrical side, the
theory of groups of contact-transformations has been developed with
very considerable detail in the second volume of Lie-Engel.

Applications of the theory of continuous groups.

To the manifold applications of the theory of continuous groups in
various branches of pure and applied mathematics it is impossible here
to refer in any detail. It must suffice to indicate a few of them very
briefly. In some of the older theories a new point of view is obtained
which presents the results in a fresh light, and suggests the natural
generalization. As an example, the theory of the invariants of a
binary form may be considered.

If in the form [f] = a0x^_n + na1x^(n-1)y + ... + a_ny^n, the
variables be subjected to a homogeneous substitution

x' = [alpha]x + [beta]y, y' = [gamma]x + [delta]y, (i.)

and if the coefficients in the new form be represented by accenting
the old coefficients, then

a'0 = a0[alpha]^n + a1n[alpha]^(n-1)[gamma] + ... + a_n[gamma]^n,\
|
a'1 = a0[alpha]^(n-1)[beta] + a1_(n-1)[alpha]^(n-2)[beta][gamma] |
+ [alpha]^(n-1)[delta]} + ... + a_n[gamma]^(n-1)[delta], > (ii.)
|
a'_n = a0[beta]^n + a1n[beta]^(n-1)[delta] + ... + a_n[delta]^n; /

and this is a homogeneous linear substitution performed on the
coefficients. The totality of the substitutions, (i.), for which
[alpha][delta] - [beta][gamma] = 1, constitutes a continuous group of
order 3, which is generated by the two infinitesimal transformations
y([Pd]/[Pd]x) and x([Pd]/[Pd]y). Hence with the same limitations on
[alpha], [beta], [gamma], [delta] the totality of the substitutions
(ii.) forms a simply isomorphic continuous group of order 3, which is
generated by the two infinitesimal transformations

[Pd] [Pd] [Pd] [Pd]
a0 ------ + 2a1 ------ + 3a1 ------ + ... + na_(n-1) -------,
[Pd]a1 [Pd]a2 [Pd]a3 [Pd]a_n

and

[Pd] [Pd] [Pd] [Pd]
na1 ------ + (n - 1)a2 ------ + (n - 2)a3 ------ + ... + a_u ----------.
[Pd]a0 [Pd]a1 [Pd]a2 [Pd]a_(u-1)

The invariants of the binary form, i.e. those functions of the
coefficients which are unaltered by all homogeneous substitutions on
x, y of determinant unity, are therefore identical with the functions
of the coefficients which are invariant for the continuous group
generated by the two infinitesimal operations last written. In other
words, they are given by the common solutions of the differential
equations

[Pd][f] [Pd][f] [Pd][f]
a0 ------- + 2a1 ------- + 3a2 ------- + ... = 0,
[Pd]a1 [Pd]a2 [Pd]a3

[Pd][f] [Pd][f] [Pd][f]
na1 ------- + (n - 1)a2 ------- + (n - 2)a3 ------- + ... = 0.
[Pd]a0 [Pd]a1 [Pd]a2

Both this result and the method by which it is arrived at are well
known, but the point of view by which we pass from the transformation
group of the variables to the isomorphic transformation group of the
coefficients, and regard the invariants as invariants rather of the
group than of the forms, is a new and a fruitful one.

The general theory of curvature of curves and surfaces may in a
similar way be regarded as a theory of their invariants for the group
of motions. That something more than a mere change of phraseology is
here implied will be evident in dealing with minimum curves, i.e. with
curves such that at every point of them dx^2 + dy^2 + dz^2 = 0. For
such curves the ordinary theory of curvature has no meaning, but they
nevertheless have invariant properties in regard to the group of
motions.

The curvature and torsion of a curve, which are invariant for all
transformations by the group of motions, are special instances of what
are known as _differential invariants_. If [xi]([Pd]/[Pd]x) +
[eta]([Pd]/[Pd]y) is the general infinitesimal transformation of a
group of point-transformations in the plane, and if y1, y2, ...
represent the successive differential coefficients of y, the
infinitesimal transformation may be written in the extended form

[Pd] [Pd] [Pd] [Pd]
[xi] ----- + [eta] ----- + [eta]1 ------ + [eta]2 ------ + ...
[Pd]x [Pd]y [Pd]y1 [Pd]y2

where [eta]1[delta]t, [eta]2[delta]t, ... are the increments of y1,
y2, .... By including a sufficient number of these variables the group
must be intransitive in them, and must therefore have one or more
invariants. Such invariants are known as differential invariants of
the original group, being necessarily functions of the differential
coefficients of the original variables. For groups of the plane it may
be shown that not more than two of these differential invariants are
independent, all others being formed from these by algebraical
processes and differentiation. For groups of point-transformations in
more than two variables there will be more than one set of
differential invariants. For instance, with three variables, one may
be regarded as independent and the other two as functions of it, or
two as independent and the remaining one as a function. Corresponding
to these two points of view, the differential invariants for a curve
or for a surface will arise.

If a differential invariant of a continuous group of the plane be
equated to zero, the resulting differential equation remains unaltered
when the variables undergo any transformation of the group.
Conversely, if an ordinary, differential equation [f](x, y, y1, y2,
...) = 0 admits the transformations of a continuous group, i.e. if the
equation is unaltered when x and y undergo any transformation of the
group, then [f](x, y, y1, y2, ...) or some multiple of it must be a
differential invariant of the group. Hence it must be possible to find
two independent differential invariants [alpha], [beta] of the group,
such that when these are taken as variables the differential equation
takes the form F([alpha], [beta], d[beta]/d[alpha],
d^2[beta]/d[alpha]^2, ...) = 0. This equation in [alpha], [beta] will
be of lower order than the original equation, and in general simpler
to deal with. Supposing it solved in the form [beta] = [phi]([alpha]),
where for [alpha], [beta] their values in terms of x, y, y1, y2, ...
are written, this new equation, containing arbitrary constants, is
necessarily again of lower order than the original equation. The
integration of the original equation is thus divided into two steps.
This will show how, in the case of an ordinary differential equation,
the fact that the equation admits a continuous group of
transformations may be taken advantage of for its integration.

The most important of the applications of continuous groups are to the
theory of systems of differential equations, both ordinary and
partial; in fact, Lie states that it was with a view to systematizing
and advancing the general theory of differential equations that he was
led to the development of the theory of continuous groups. It is quite
impossible here to give any account of all that Lie and his followers
have done in this direction. An entirely new mode of regarding the
problem of the integration of a differential equation has been opened
up, and in the classification that arises from it all those apparently
isolated types of equations which in the older sense are said to be
integrable take their proper place. It may, for instance, be mentioned
that the question as to whether Monge's method will apply to the
integration of a partial differential equation of the second order is
shown to depend on whether or not a contact-transformation can be
found which will reduce the equation to either [Pd]^2z/[Pd]x^2 = 0 or
[Pd]^2z/[Pd]x[Pd]y = 0. It is in this direction that further advance
in the theory of partial differential equations must be looked for.
Lastly, it may be remarked that one of the most thorough discussions
of the axioms of geometry hitherto undertaken is founded entirely upon
the theory of continuous groups.

_Discontinuous Groups._

We go on now to the consideration of discontinuous groups. Although groups of finite order are necessarily contained under this general head, it is convenient for many reasons to deal with them separately, and it will therefore be assumed in the present section that the number of operations in the group is not finite. Many large classes of discontinuous groups have formed the subject of detailed investigation, but a general formal theory of discontinuous groups can hardly be said to exist as yet. It will thus be obvious that in considering discontinuous groups it is necessary to proceed on different lines from those followed with continuous groups, and in fact to deal with the subject almost entirely by way of example.

Generating operations.

The consideration of a discontinuous group as arising from a set of
independent generating operations suggests a purely abstract point of
view in which any two simply isomorphic groups are indistinguishable.
The number of generating operations may be either finite or infinite,
but the former case alone will be here considered. Suppose then that
S1, S2, ..., S_n is a set of independent operations from which a group
G is generated. The general operation of the group will be represented
by the symbol S_a^[alpha]S_b^[beta] ... S_d^[delta], or [Sigma], where
a, b, ..., d are chosen from 1, 2, ..., n, and [alpha], [beta], ...,
[delta] are any positive or negative integers. It may be assumed that
no two successive suffixes in [Sigma] are the same, for if b = a, then
S_a^[alpha]S_b^[beta] may be replaced by S_a^([alpha] +[beta]). If
there are no relations connecting the generating operations and the
identical operation, every distinct symbol [Sigma] represents a
distinct operation of the group. For if [Sigma] = [Sigma]1, or
S_a^[alpha] S_b^[beta] ... S_d^[delta] = S_(a1)^([alpha]1)
S_(b1)^([beta]1) ... S_(d1)^([delta]1), then S_(d1)^(-[delta]1) ...
S_(b1)^(-[beta]1) S_(a1)^(-[alpha]1) S_a^[alpha] S_b^[beta] ...
S_d^[delta] = 1; and unless a = a1, b = b1, ..., [alpha] = [alpha]1,
[beta] = [beta]1, ..., this is a relation connecting the generating
operations.

Suppose now that T1, T2, ... are operations of G, and that H is that
self-conjugate subgroup of G which is generated by T1, T2, ... and the
operations conjugate to them. Then, of the operations that can be
formed from S1, S2, ..., S_n, the set [Sigma]H, and no others, reduce
to the same operation [Sigma] when the conditions T1 = 1, T2 = 1, ...
are satisfied by the generating operations. Hence the group which is
generated by the given operations, when subjected to the conditions
just written, is simply isomorphic with the factor-group G/H.
Moreover, this is obviously true even when the conditions are such
that the generating operations are no longer independent. Hence any
discontinuous group may be defined abstractly, that is, in regard to
the laws of combination of its operations apart from their actual
form, by a set of generating operations and a system of relations
connecting them. Conversely, when such a set of operations and system
of relations are given arbitrarily they define in abstract form a
single discontinuous group. It may, of course, happen that the group
so defined is a group of finite order, or that it reduces to the
identical operation only; but in regard to the general statement these
will be particular and exceptional cases.

Properly and improperly discontinuous groups.

An operation of a discontinuous group must necessarily be specified
analytically by a system of equations of the form

x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n),

and the different operations of the group will be given by different
sets of values of the parameters a1, a2, ..., a_r. No one of these
parameters is susceptible of continuous variations, but at least one
must be capable of taking a number of values which is not finite, if
the group is not one of finite order. Among the sets of values of the
parameters there must be one which gives the identical transformation.
No other transformation makes each of the differences x'1 - x1, x'2 -
x2, ..., x'_n - x_n vanish. Let d be an arbitrary assigned positive
quantity. Then if a transformation of the group can be found such that
the modulus of each of these differences is less than d when the
variables have arbitrary values within an assigned range of variation,
however small d may be chosen, the group is said to be _improperly_
discontinuous. In the contrary case the group is called _properly_
discontinuous. The range within which the variables are allowed to
vary may clearly affect the question whether a given group is properly
or improperly discontinuous. For instance, the group defined by the
equation x' = ax + b, where a and b are any rational numbers, is
improperly discontinuous; and the group defined by x' = x + a, where a
is an integer, is properly discontinuous, whatever the range of the
variable. On the other hand, the group, to be later considered,
defined by the equation x' = (ax + b)/(cx + d), where a, b, c, d are
integers satisfying the relation ad - bc = 1, is properly
discontinuous when x may take any complex value, and improperly
discontinuous when the range of x is limited to real values.

Linear discontinuous groups.

Among the discontinuous groups that occur in analysis, a large number
may be regarded as arising by imposing limitations on the range of
variation of the parameters of continuous groups. If

x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n),

are the finite equations of a continuous group, and if C with
parameters c1, c2, ..., c_r is the operation which results from
carrying out A and B with corresponding parameters in succession, then
the c's are determined uniquely by the a's and the b's. If the c's are
rational functions of the a's and b's, and if the a's and b's are
arbitrary rational numbers of a given corpus (see NUMBER), the c's
will be rational numbers of the same corpus. If the c's are rational
integral functions of the a's and b's, and the latter are arbitrarily
chosen integers of a corpus, then the c's are integers of the same
corpus. Hence in the first case the above equations, when the a's are
limited to be rational numbers of a given corpus, will define a
discontinuous group; and in the second case they will define such a
group when the a's are further limited to be integers of the corpus. A
most important class of discontinuous groups are those that arise in
this way from the general linear continuous group in a given set of
variables. For n variables the finite equations of this continuous
group are

x'_s = a_(s1)x1 + a_(s2)x2 + ... + a_(sn)x_n, (s = 1, 2, ..., n),

where the determinant of the a's must not be zero. In this case the
c's are clearly integral lineo-linear functions of the a's and b's.
Moreover, the determinant of the c's is the product of the determinant
of the a's and the determinant of the b's. Hence equations (ii.),
where the parameters are restricted to be integers of a given corpus,
define a discontinuous group; and if the determinant of the
coefficients is limited to the value unity, they define a
discontinuous group which is a (self-conjugate) subgroup of the
previous one.

The simplest case which thus presents itself is that in which there
are two variables while the coefficients are rational integers. This
is the group defined by the equations

x' = ax + by, \
>
y' = cx + dy, /

where a, b, c, d are integers such that ad - bc = 1. To every
operation of this group there corresponds an operation of the set
defined by

az + b
z' = ------,
cz + d

in such a way that to the product of two operations of the group there
corresponds the product of the two analogous operations of the set.
The operations of the set (iv.), where ad - bc = 1, therefore
constitute a group which is isomorphic with the previous group. The
isomorphism is multiple, since to a single operation of the second set
there correspond the two operations of the first for which a, b, c, d
and -a, -b, -c, -d are parameters. These two groups, which are of
fundamental importance in the theory of quadratic forms and in the
theory of modular functions, have been the object of very many
investigations.

Discontinuous groups arising from geometrical operations.

Another large class of discontinuous groups, which have far-reaching
applications in analysis, are those which arise in the first instance
from purely geometrical considerations. By the combination and
repetition of a finite number of geometrical operations such as
displacements, projective transformations, inversions, &c., a
discontinuous group of such operations will arise. Such a group, as
regards the points of the plane (or of space), will in general be
improperly discontinuous; but when the generating operations are
suitably chosen, the group may be properly discontinuous. In the
latter case the group may be represented in a graphical form by the
division of the plane (or space) into regions such that no point of
one region can be transformed into another point of the same region by
any operation of the group, while any given region can be transformed
into any other by a suitable transformation. Thus, let ABC be a
triangle bounded by three circular arcs BC, CA, AB; and consider the
figure produced from ABC by inversions in the three circles of which
BC, CA, AB are part. By inversion at BC, ABC becomes an equiangular
triangle A'BC. An inversion in AB changes ABC and A'BC into
equiangular triangles ABC' and A"BC'. Successive inversions at AB and
BC then will change ABC into a series of equiangular triangles with B
for a common vertex. These will not overlap and will just fill in the
space round B if the angle ABC is a submultiple of two right angles.
If then the angles of ABC are submultiples of two right angles (or
zero), the triangles formed by any number of inversions will never
overlap, and to each operation consisting of a definite series of
inversions at BC, CA and AB will correspond a distinct triangle into
which ABC is changed by the operation. The network of triangles so
formed gives a graphical representation of the group that arises from
the three inversions in BC, CA, AB. The triangles may be divided into
two sets, those, namely, like A"BC', which are derived from ABC by an
even number of inversions, and those like A'BC or ABC' produced by an
odd number. Each set are interchanged among themselves by any even
number of inversions. Hence the operations consisting of an even
number of inversions form a group by themselves. For this group the
quadrilateral formed by ABC and A'BC constitutes a region, which is
changed by every operation of the group into a distinct region (formed
of two adjacent triangles), and these regions clearly do not overlap.
Their distribution presents in a graphical form the group that arises
by pairs of inversions at BC, CA, AB; and this group is generated by
the operation which consists of successive inversions at AB, BC and
that which consists of successive inversions at BC, CA. The group
defined thus geometrically may be presented in many analytical forms.
If x, y and x', y' are the rectangular co-ordinates of two points
which are inverse to each other with respect to a given circle, x' and
y' are rational functions of x and y, and conversely. Thus the group
may be presented in a form in which each operation gives a birational
transformation of two variables. If x + iy = z, x' + iy' = z', and if
x', y' is the point to which x, y is transformed by any even number of
inversions, then z' and z are connected by a linear relation z' =
([alpha]z + [beta])/([gamma]z + [delta]), where [alpha], [beta],
[gamma], [delta] are constants (in general complex) depending on the
circles at which the inversions are taken. Hence the group may be
presented in the form of a group of linear transformations of a single
variable generated by the two linear transformations z' = ([alpha]1z +
[beta]1)/([gamma]1z + [delta]1), z' = ([alpha]2z + [beta]2)/([gamma]2z
+ [delta]2), which correspond to pairs of inversions at AB, BC and BC,
CA respectively. In particular, if the sides of the triangle are taken
to be x = 0, x^2 + y^2 -1 = 0, x^2 + y^2 + 2x = 0, the generating
operations are found to be z' = z + 1, z' = -z^(-1); and the group is
that consisting of all transformations of the form z' = (az + b)/(cz +
d), where ad - bc = 1, a, b, c, d being integers. This is the group
already mentioned which underlies the theory of the elliptic modular
functions; a modular function being a function of z which is invariant
for some subgroup of finite index of the group in question.

The triangle ABC from which the above geometrical construction started
may be replaced by a polygon whose sides are circles. If each angle is
a submultiple of two right angles or zero, the construction is still
effective to give a set of non-overlapping regions, which represent
graphically the group which arises from pairs of inversions in the
sides of the polygon. In their analytical form, as groups of linear
transformations of a single variable, the groups are those on which
the theory of automorphic functions depends. A similar construction in
space, the polygons bounded by circular arcs being replaced by
polyhedra bounded by spherical faces, has been used by F. Klein and
Fricke to give a geometrical representation for groups which are
improperly discontinuous when represented as groups of the plane.

Group of a linear differential equation.

The special classes of discontinuous groups that have been dealt with
in the previous paragraphs arise directly from geometrical
considerations. As a final example we shall refer briefly to a class
of groups whose origin is essentially analytical. Let

d^_ny d^(n-1)y dy
----- + P1 -------- + ... + P_(n-1) -- + P_ny = 0
dx^_n dx^(n-1) dx

be a linear differential equation, the coefficients in which are
rational functions of x, and let y1, y2, ..., y_n be a linearly
independent set of integrals of the equation. In the neighbourhood of
a finite value x0 of x, which is not a singularity of any of the
coefficients in the equation, these integrals are ordinary
power-series in x - x0. If the analytical continuations of y1, y2,
..., y_n be formed for any closed path starting from and returning to
x0, the final values arrived at when x0 is again reached will be
another set of linearly independent integrals. When the closed path
contains no singular point of the coefficients of the differential
equation, the new set of integrals is identical with the original set.
If, however, the closed path encloses one or more singular points,
this will not in general be the case. Let y'1, y'2, ..., y'_n be the
new integrals arrived at. Since in the neighbourhood of x0 every
integral can be represented linearly in terms of y1, y2, ..., y_n,
there must be a system of equations

y'1 = a11y1 + a12y2 + ... + a_(1n)y_n,

y'2 = a21y1 + a22y2 + ... + a_(2n)y_n,

. . . . .

y'_n = a_(n1)y1 + a_(n2)y2 + ... + a_(nn)y_n,

where the a's are constants, expressing the new integrals in terms of
the original ones. To each closed path described by x0 there therefore
corresponds a definite linear substitution performed on the y's.
Further, if S1 and S2 are the substitutions that correspond to two
closed paths L1 and L2, then to any closed path which can be
continuously deformed, without crossing a singular point, into L1
followed by L2, there corresponds the substitution S1S2. Let L1, L2,
..., L_r be arbitrarily chosen closed paths starting from and
returning to the same point, and each of them enclosing a single one
of the (r) finite singular points of the equation. Every closed path
in the plane can be formed by combinations of these r paths taken
either in the positive or in the negative direction. Also a closed
path which does not cut itself, and encloses all the r singular points
within it, is equivalent to a path enclosing the point at infinity and
no finite singular point. If S1, S2, S3, ..., S_r are the linear
substitutions that correspond to these r paths, then the substitution
corresponding to every possible path can be obtained by combination
and repetition of these r substitutions, and they therefore generate a
discontinuous group each of whose operations corresponds to a definite
closed path. The group thus arrived at is called the group of the
equation. For a given equation it is unique in type. In fact, the only
effect of starting from another set of independent integrals is to
transform every operation of the group by an arbitrary substitution,
while choosing a different set of paths is equivalent to taking a new
set of generating operations. The great importance of the group of the
equation in connexion with the nature of its integrals cannot here be
dealt with, but it may be pointed out that if all the integrals of the
equation are algebraic functions, the group must be a group of finite
order, since the set of quantities y1, y2 ..., y_n can then only take
a finite number of distinct values.

_Groups of Finite Order._

We shall now pass on to groups of finite order. It is clear that here we must have to do with many properties which have no direct analogues in the theory of continuous groups or in that of discontinuous groups in general; those properties, namely, which depend on the fact that the number of distinct operations in the group is finite.

Let S1, S2, S3, ..., S_N denote the operations of a group G of finite
order N, S1 being the identical operation. The tableau

S1, S2, S3, ..., S_N,
S1S2, S2S2, S3S3, ..., S_NS2,
S1S3, S2S3, S3S3, ..., S_NS3,
. . . . .
S1S_N, S2S_N, S3S_N, ..., S_NS_N,

when in it each compound symbol S_pS_q is replaced by the single
symbol S_r that is equivalent to it, is called the multiplication
table of the group. It indicates directly the result of multiplying
together in an assigned sequence any number of operations of the
group. In each line (and in each column) of the tableau every
operation of the group occurs just once. If the letters in the tableau
are regarded as mere symbols, the operation of replacing each symbol
in the first line by the symbol which stands under it in the pth line
is a permutation performed on the set of N symbols. Thus to the N
lines of the tableau there corresponds a set of N permutations
performed on the N symbols, which includes the identical permutation
that leaves each unchanged. Moreover, if S_pS_q = S_r, then the result
of carrying out in succession the permutations which correspond to the
pth and qth lines gives the permutation which corresponds to the rth
line. Hence the set of permutations constitutes a group which is
simply isomorphic with the given group.

Every group of finite order N can therefore be represented in concrete
form as a transitive group of permutations on N symbols.

Properties of a group which depend on the order.

The order of any subgroup or operation of G is necessarily finite. If
T1(= S1), T2, ..., T_n are the operations of a subgroup H of G, and if
[Sigma] is any operation of G which is not contained in H, the set of
operations [Sigma]T1, [Sigma]T2, ..., [Sigma]T_n, or [Sigma]H, are all
distinct from each other and from the operations of H. If the sets H
and [Sigma]H do not exhaust the operations of G, and if [Sigma]' is an
operation not belonging to them, then the operations of the set
[Sigma]'H are distinct from each other and from those of H and
[Sigma]H. This process may be continued till the operations of G are
exhausted. The order n of H must therefore be a factor of the order N
of G. The ratio N/n is called the index of the subgroup H. By taking
for H the cyclical subgroup generated by any operation S of G, it
follows that the order of S must be a factor of the order of G.

Every operation S is permutable with its own powers. Hence there must
be some subgroup H of G of greatest possible order, such that every
operation of H is permutable with S. Every operation of H transforms S
into itself, and every operation of the set H[Sigma] transforms S into
the same operation. Hence, when S is transformed by every operation of
G, just N/n distinct operations arise if n is the order of H. These
operations, and no others, are conjugate to S within G; they are said
to form a set of conjugate operations. The number of operations in
every conjugate set is therefore a factor of the order of G. In the
same way it may be shown that the number of subgroups which are
conjugate to a given subgroup is a factor of the order of G. An
operation which is permutable with every operation of the group is
called a _self-conjugate_ operation. The totality of the
self-conjugate operations of a group forms a self-conjugate Abelian
subgroup, each of whose operations is permutable with every operation
of the group.

Sylow's theorem.

An Abelian group contains subgroups whose orders are any given factors
of the order of the group. In fact, since every subgroup H of an
Abelian group G and the corresponding factor groups G/H are Abelian,
this result follows immediately by an induction from the case in which
the order contains n prime factors to that in which it contains n + 1.
For a group which is not Abelian no general law can be stated as to
the existence or non-existence of a subgroup whose order is an
arbitrarily assigned factor of the order of the group. In this
connexion the most important general result, which is independent of
any supposition as to the order of the group, is known as Sylow's
theorem, which states that if p^a is the highest power of a prime p
which divides the order of a group G, then G contains a single
conjugate set of subgroups of order p^a, the number in the set being
of the form 1 + kp. Sylow's theorem may be extended to show that if
p^a' is a factor of the order of a group, the number of subgroups of
order p^a' is of the form 1 + kp. If, however, p^a' is not the highest
power of p which divides the order, these groups do not in general
form a single conjugate set.

The importance of Sylow's theorem in discussing the structure of a
group of given order need hardly be insisted on. Thus, as a very
simple instance, a group whose order is the product p1p2 of two primes
(p1 < p2) must have a self-conjugate subgroup of order p2, since the
order of the group contains no factor, other than unity, of the form 1
+ kp2. The same again is true for a group of order p1^2p2, unless p1 =
2, and p2 = 3.

There is one other numerical property of a group connected with its
order which is quite general. If N is the order of G, and n a factor
of N, the number of operations of G, whose orders are equal to or are
factors of n, is a multiple of n.

Composition-series of a group.

As already defined, a composite group is a group which contains one or
more self-conjugate subgroups, whose orders are greater than unity. If
H is a self-conjugate subgroup of G, the factor-group G/H may be
either simple or composite. In the former case G can contain no
self-conjugate subgroup K, which itself contains H; for if it did K/H
would be a self-conjugate subgroup of G/H. When G/H is simple, H is
said to be a maximum self-conjugate subgroup of G. Suppose now that G
being a given composite group, G, G1, G2, ..., G_n, 1 is a series of
subgroups of G, such that each is a maximum self-conjugate subgroup of
the preceding; the last term of the series consisting of the identical
operation only. Such a series is called a _composition-series_ of G.
In general it is not unique, since a group may have two or more
maximum self-conjugate subgroups. A composition-series of a group,
however it may be chosen, has the property that the number of terms of
which it consists is always the same, while the factor-groups G/G1,
G1/G2, ..., G_n differ only in the sequence in which they occur. It
should be noticed that though a group defines uniquely the set of
factor-groups that occur in its composition-series, the set of
factor-groups do not conversely in general define a single type of
group. When the orders of all the factor-groups are primes the group
is said to be _soluble_.

If the series of subgroups G, H, K, ..., L, 1 is chosen so that each
is the greatest self-conjugate subgroup of G contained in the previous
one, the series is called a chief composition-series of G. All such
series derived from a given group may be shown to consist of the same
number of terms, and to give rise to the same set of factor-groups,
except as regards sequence. The factor-groups of such a series will
not, however, necessarily be simple groups. From any chief
composition-series a composition-series may be formed by interpolating
between any two terms H and K of the series for which H/K is not a
simple group, a number of terms h1, h2, ..., h_r; and it may be shown
that the factor-groups H/h1, h1/h2, ..., h_r/K are all simply
isomorphic with each other.

Isomorphism of a group with itself.

A group may be represented as isomorphic with itself by transforming
all its operations by any one of them. In fact, if S_pS_q = S_r, then
S^(-1)S_pS . S^(-1)S_qS = S^(-1)S_rS. An isomorphism of the group with
itself, established in this way, is called an inner isomorphism. It
may be regarded as an operation carried out on the symbols of the
operations, being indeed a permutation performed on these symbols. The
totality of these operations clearly constitutes a group isomorphic
with the given group, and this group is called the group of inner
isomorphisms. A group is simply or multiply isomorphic with its group
of inner isomorphisms according as it does not or does contain
self-conjugate operations other than identity. It may be possible to
establish a correspondence between the operations of a group other
than those given by the inner isomorphisms, such that if S' is the
operation corresponding to S, then S'_pS'_q = S'_r is a consequence of
S_pS_q = S_r. The substitution on the symbols of the operations of a
group resulting from such a correspondence is called an outer
isomorphism. The totality of the isomorphisms of both kinds
constitutes the group of isomorphisms of the given group, and within
this the group of inner isomorphisms is a self-conjugate subgroup.
Every set of conjugate operations of a group is necessarily
transformed into itself by an inner isomorphism, but two or more sets
may be interchanged by an outer isomorphism.

A subgroup of a group G, which is transformed into itself by every
isomorphism of G, is called a _characteristic_ subgroup. A series of
groups G, G1, G2, ..., 1, such that each is a maximum characteristic
subgroup of G contained in the preceding, may be shown to have the
same invariant properties as the subgroups of a composition series. A
group which has no characteristic subgroup must be either a simple
group or the direct product of a number of simply isomorphic simple
groups.

Permutation-groups.

It has been seen that every group of finite order can be represented
as a group of permutations performed on a set of symbols whose number
is equal to the order of the group. In general such a representation
is possible with a smaller number of symbols. Let H be a subgroup of
G, and let the operations of G be divided, in respect of H, into the
sets H, S2H, S3H, ..., S_mH. If S is any operation of G, the sets SH,
SS2H, SS3H, ..., SS_mH differ from the previous sets only in the
sequence in which they occur. In fact, if SS_p belong to the set S_qH,
then since H is a group, the set SS_pH is identical with the set S_qH.
Hence, to each operation S of the group will correspond a permutation
performed on the symbols of the m sets, and to the product of two
operations corresponds the product of the two analogous permutations.
The set of permutations, therefore, forms a group isomorphic with the
given group. Moreover, the isomorphism is simple unless for one or
more operations, other than identity, the sets all remain unaltered.
This can only be the case for S, when every operation conjugate to S
belongs to H. In this case H would contain a self-conjugate subgroup,
and the isomorphism is multiple.

The fact that every group of finite order can be represented,
generally in several ways, as a group of permutations, gives special
importance to such groups. The number of symbols involved in such a
representation is called the _degree_ of the group. In accordance with
the general definitions already given, a permutation-group is called
transitive or intransitive according as it does or does not contain
permutations changing any one of the symbols into any other. It is
called imprimitive or primitive according as the symbols can or cannot
be arranged in sets, such that every permutation of the group changes
the symbols of any one set either among themselves or into the symbols
of another set. When a group is imprimitive the number of symbols in
each set must clearly be the same.

The total number of permutations that can be performed on n symbols is
n!, and these necessarily constitute a group. It is known as the
_symmetric_ group of degree n, the only rational functions of the
symbols which are unaltered by all possible permutations being the
symmetric functions. When any permutation is carried out on the
product of the n(n - 1)/2, differences of the n symbols, it must
either remain unaltered or its sign must be changed. Those
permutations which leave the product unaltered constitute a group of
order n!/2, which is called the _alternating_ group of degree n; it is
a self-conjugate subgroup of the symmetric group. Except when n = 4
the alternating group is a simple group. A group of degree n, which is
not contained in the alternating group, must necessarily have a
self-conjugate subgroup of index 2, consisting of those of its
permutations which belong to the alternating group.

Groups of linear substitutions.

Among the various concrete forms in which a group of finite order can
be presented the most important is that of a group of linear
substitutions. Such groups have already been referred to in connexion
with discontinuous groups. Here the number of distinct substitutions
is necessarily finite; and to each operation S of a group G of finite
order there will correspond a linear substitution s, viz.

__j=m
x_i = \ s_(ij)x_j(i, j = 1, 2, ..., m),
/__j=1

on a set of m variables, such that if ST = U, then st = u. The linear
substitutions s, t, u, ... then constitute a group g with which G is
isomorphic; and whether the isomorphism is simple or multiple g is
said to give a "representation" of G as a group of linear
substitutions. If all the substitutions of g are transformed by the
same substitution on the m variables, the (in general) new group of
linear substitutions so constituted is said to be "equivalent" with g
as a representation of G; and two representations are called
"non-equivalent," or "distinct," when one is not capable of being
transformed into the other.

A group of linear substitutions on m variables is said to be
"reducible" when it is possible to choose m'(< m) linear functions of
the variables which are transformed among themselves by every
substitution of the group. When this cannot be done the group is
called "irreducible." It can be shown that a group of linear
substitutions, of finite order, is always either irreducible, or such
that the variables, when suitably chosen, may be divided into sets,
each set being irreducibly transformed among themselves. This being
so, it is clear that when the irreducible representations of a group
of finite order are known, all representations may be built up.

It has been seen at the beginning of this section that every group of
finite order N can be presented as a group of permutations (i.e.
linear substitutions in a limited sense) on N symbols. This group is
obviously reducible; in fact, the sum of the symbols remain unaltered
by every substitution of the group. The fundamental theorem in
connexion with the representations, as an irreducible group of linear
substitutions, of a group of finite order N is the following.

If r is the number of different sets of conjugate operations in the
group, then, when the group of N permutations is completely reduced,

(i.) just r distinct irreducible representations occur:

(ii.) each of these occurs a number of times equal to the number of
symbols on which it operates:

(iii.) these irreducible representations exhaust all the distinct
irreducible representations of the group.

Among these representations what is called the "identical"
representation necessarily occurs, i.e. that in which each operation
of the group corresponds to leaving a single symbol unchanged. If
these representations are denoted by [Gamma]1, [Gamma]2, ...,
[Gamma]_r, then any representation of the group as a group of linear
substitutions, or in particular as a group of permutations, may be
uniquely represented by a symbol [Sigma][alpha]_i[Gamma]_i, in the
sense that the representation when completely reduced will contain the
representation [Gamma]_i just [alpha]_i times for each suffix i.

Group characteristics.

A representation of a group of finite order as an irreducible group of
linear substitutions may be presented in an infinite number of
equivalent forms. If

x'_i = [Sigma] s_(ij)x_j (i, j = 1, 2, ..., m),

is the linear substitution which, in a given irreducible
representation of a group of finite order G, corresponds to the
operation S, the determinant

| s11 - [lambda] s12 ... s_(1m) |
| s21 s22-[lambda] ... s_(2m) |
| . . ... . |
| . . ... . |
| . . ... . |
| s_m1 s_2m ... s_(mm) - [lambda] |