Chapter I: Front Matter (1)

Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
paragraphs.

(4) Macrons and breves above letters and dots below letters were not
inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
letters and [Pd] for partial differential symbol.

(6) The following typographical errors have been corrected:

ARTICLE GUADALQUIVIR: "Here it forms two subsidiary channels, the
western 31 m., the eastern 12 m. long, which rejoin the main stream
on the borders of the province of Cadiz." 'm.' amended from 'M.'.

ARTICLE GUANAJUATO: "... W. of Guanajuato in a rich mining
district; and Acambaro (8345), a prosperous town of the plain, 76
m. S.S.E. of Guanajuato." 'Guanajuato' amended from 'Guanaiuato'.

ARTICLE GUARANTEE: "The Egyptian codes sanction guarantees
expressly entered into 'in view of debtor's want of legal capacity'
to contract a valid principal obligation (Egyptian Codes, Mixed
Suits, 605; Native Tribunals, 496)." 'Egyptian' amended from
'Egyptain'.

ARTICLE GUINEA FOWL: "Allied to the genus Numida, but readily
distinguished form among other characters by the possession of
spurs and the absence of a helmet, are two very rare forms ..."
'form' amended from 'thereform'.

ARTICLE GUIPUZCOA: "The principal industrial centres are Irun,
Renteria, Villabona, Vergara and Azpeitia for cotton and linen
stuffs; Zumarraga for osiers; Eibar, Plasencia and Elgoibar for
arms and cannon and gold incrustations; ..." 'osiers' amended from
'osies'.

ARTICLE GUTZKOW, KARL FERDINAND: "The success of Die Ritter vom
Geiste suggested to Gutzkow the establishment of a journal on the
model of Dickens' Household Words, entitled Unterhaltungen am
hauslichen Herd, which first appeared in 1852 and was continued
till 1862." "Dickens'" amended from "Dicken's".

ARTICLE GUY OF WARWICK: "... The Tragical History, Admirable
Achievements and Curious Events of Guy, Earl of Warwick ..."
'Achievements' amended from 'Atchievements'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME XII, SLICE VI

Groups, Theory of to Gwyniad

ARTICLES IN THIS SLICE:

GROUPS, THEORY OF GUIDICCIONI, GIOVANNI
GROUSE GUIDO OF AREZZO
GROVE, SIR GEORGE GUIDO OF SIENA
GROVE, SIR WILLIAM ROBERT GUIDO RENI
GROVE GUIENNE
GROZNYI GUIGNES, JOSEPH DE
GRUB GUILBERT, YVETTE
GRUBER, JOHANN GOTTFRIED GUILDFORD
GRUMBACH, WILHELM VON GUILDHALL
GRUMENTUM GUILFORD, BARONS AND EARLS OF
GRUN GUILFORD
GRUNBERG GUILLAUME, JEAN BAPTISTE CLAUDE EUGENE
GRUNDTVIG, NIKOLAI SEVERIN GUILLAUME DE LORRIS
GRUNDY, SYDNEY GUILLAUME DE PALERME
GRUNDY, MRS GUILLAUME D'ORANGE
GRUNER, GOTTLIEB SIGMUND GUILLEMOT
GRUNEWALD, MATHIAS GUILLOCHE
GRUTER, JAN GUILLON, MARIE NICOLAS SYLVESTRE
GRUYERE GUILLOTINE
GRYNAEUS, JOHANN JAKOB GUILT
GRYNAEUS, SIMON GUIMARAES
GRYPHIUS, ANDREAS GUIMARD, MARIE MADELEINE
GUACHARO GUIMET, JEAN BAPTISTE
GUACO GUINEA (Africa)
GUADALAJARA (city of Mexico) GUINEA (gold coin)
GUADALAJARA (province of Spain) GUINEA FOWL
GUADALAJARA (city of Spain) GUINEA-WORM
GUADALQUIVIR GUINES
GUADELOUPE GUINGAMP
GUADET, MARGUERITE ELIE GUINNESS
GUADIANA GUINOBATAN
GUADIX GUIPUZCOA
GUADUAS GUIRAUD, ERNEST
GUAIACUM GUISBOROUGH
GUALDO TADINO GUISE
GUALEGUAY GUISE, HOUSE OF
GUALEGUAYCHU GUITAR
GUALO, CARDINAL GUITAR FIDDLE
GUAM GUITRY, LUCIEN GERMAIN
GUAN GUIZOT, FRANCOIS PIERRE GUILLAUME
GUANABACOA GUJARAT
GUANACO GUJARATI and RAJASTHANI
GUANAJAY GUJRANWALA
GUANAJUATO (state of Mexico) GUJRAT
GUANAJUATO (city of Mexico) GULA
GUANCHES GULBARGA
GUANIDINE GULF STREAM
GUANO GULFWEED
GUANTA GULL, SIR WILLIAM WITHEY
GUANTANAMO GULL
GUARANA GULLY, JOHN
GUARANIS GULPAIGAN
GUARANTEE GUM
GUARATINGUETA GUMBEL, KARL WILHELM VON
GUARDA GUMBINNEN
GUARDI, FRANCESCO GUMBO
GUARDIAN GUMTI
GUARDS, and HOUSEHOLD TROOPS GUMULJINA
GUARD-SHIP GUMUS
GUARICO GUMUSH-KHANEH
GUARIENTO GUN
GUARINI, CAMILLO-GUARINO GUNA
GUARINI, GIOVANNI BATTISTA GUNCOTTON
GUARINO GUNDULICH, IVAN
GUARINO [GUARINUS] DA VERONA GUNG'L, JOSEF
GUARNIERI GUNNER
GUASTALLA GUNNING, PETER
GUATEMALA (republic) GUNNY
GUATEMALA (city of Guatemala) GUNPOWDER
GUATOS GUNPOWDER PLOT
GUATUSOS GUN-ROOM
GUAVA GUNTER, EDMUND
GUAYAMA GUNTHER, JOHANN CHRISTIAN
GUAYAQUIL GUNTHER OF SCHWARZBURG
GUAYAS GUNTRAM
GUAYCURUS GUNTUR
GUAYMAS GUPTA
GUBBIO GURA, EUGEN
GUBEN GURDASPUR
GUBERNATIS, ANGELO DE GURGAON
GUDBRANDSDAL GURKHA
GUDE, MARQUARD GURNALL, WILLIAM
GUDEMAN, ALFRED GURNARD
GUDGEON GURNEY
GUDRUN GURNEY, EDMUND
GUEBRIANT, JEAN BAPTISTE BUDES GURWOOD, JOHN
GUELDER ROSE GUSLA
GUELPH GUSTAVUS I. ERIKSSON
GUELPHS AND GHIBELLINES GUSTAVUS II. ADOLPHUS
GUENEVERE GUSTAVUS III.
GUENON GUSTAVUS IV.
GUERET GUSTAVUS V.
GUEREZA GUSTAVUS ADOLPHUS UNION
GUERICKE, HEINRICH FERDINAND GUSTROW
GUERICKE, OTTO VON GUTENBERG, JOHANN
GUERIDON GUTERSLOH
GUERIN, JEAN BAPTISTE PAULIN GUTHRIE, SIR JAMES
GUERIN, PIERRE NARCISSE GUTHRIE, THOMAS
GUERIN DU CAYLA, MAURICE DE GUTHRIE, THOMAS ANSTEY
GUERNIERI GUTHRIE
GUERNSEY GUTHRUM
GUERRAZZI, FRANCESCO DOMENICO GUTSCHMID, ALFRED
GUERRERO GUTS-MUTHS, JOHANN CHRISTOPH FRIEDRICH
GUERRILLA GUTTA
GUERRINI, OLINDO GUTTA PERCHA
GUESDE, JULES BASILE GUTTER
GUEST, EDWIN GUTZKOW, KARL FERDINAND
GUEST GUTZLAFF, KARL FRIEDRICH AUGUST
GUETTARD, JEAN ETIENNE GUY OF WARWICK
GUEUX, LES GUY, THOMAS
GUEVARA, ANTONIO DE GUYON, JEANNE BOUVIER DE LA MOTHE
GUEVARA, LUIS VELEZ DE GUYON, RICHARD DEBAUFRE
GUGLIELMI, PIETRO GUYOT, ARNOLD HENRY
GUIANA GUYOT, YVES
GUIART, GUILLAUME GUYTON DE MORVEAU, LOUIS BERNARD
GUIBERT (of Ravenna) GUZMICS, IZIDOR
GUIBERT (of Nogent) GWADAR
GUIBERT, JACQUES HIPPOLYTE GWALIOR
GUICCIARDINI, FRANCESCO GWEEDORE
GUICHARD, KARL GOTTLIEB GWILT, JOSEPH
GUICHEN, LUC URBAIN DE BOUEXIC GWYN, NELL
GUIDE GWYNIAD
GUIDI, CARLO ALESSANDRO

GROUPS,[1] THEORY OF. The conception of an operation to be carried out on some object or set of objects underlies all mathematical science. Thus in elementary arithmetic there are the fundamental operations of the addition and the multiplication of integers; in algebra a linear transformation is an operation which may be carried out on any set of variables; while in geometry a translation, a rotation, or a projective transformation are operations which may be carried out on any figure.

In speaking of an operation, an object or a set of objects to which it may be applied is postulated; and the operation may, and generally will, have no meaning except in regard to such a set of objects. If two operations, which can be performed on the same set of objects, are such that, when carried out in succession on any possible object, the result, whichever operation is performed first, is to produce no change in the object, then each of the operations is spoken of as a _definite_ operation, and each of them is called the _inverse_ of the other. Thus the operations which consist in replacing x by nx and by x/n respectively, in any rational function of x, are definite inverse operations, if n is any assigned number except zero. On the contrary, the operation of replacing x by an assigned number in any rational function of x is not, in the present sense, although it leads to a unique result, a definite operation; there is in fact no unique inverse operation corresponding to it. It is to be noticed that the question whether an operation is a definite operation or no may depend on the range of the objects on which it operates. For example, the operations of squaring and extracting the square root are definite inverse operations if the objects are restricted to be real positive numbers, but not otherwise.

If O, O', O", ... is the totality of the objects on which a definite
operation S and its inverse S' may be carried out, and if the result
of carrying out S on O is represented by O.S, then O.S.S', O.S'.S, and
O are the same object whatever object of the set O may be. This will
be represented by the equations SS' = S'S = 1. Now O.S.S' has a
meaning only if O.S is an object on which S' may be performed. Hence
whatever object of the set O may be, both O.S and O.S' belong to the
set. Similarly O.S.S, O.S.S.S, ... are objects of the set. These will
be represented by O.S^2, O.S^3, ... Suppose now that T is another
definite operation with the same set of objects as S, and that T' is
its inverse operation. Then O.S.T is a definite operation of the set,
and therefore the result of carrying out S and then T on the set of
objects is some operation U with a unique result. Represent by U' the
result of carrying out T' and then S'. Then O.UU' = O.S.T.T'.S' =
O.SS' = O, and O.U'U = O.T'.S'.S.T = O.T'T = O, whatever object O may
be. Hence UU' = U'U = 1; and U, U' are definite inverse operations.

If S, U, V are definite operations, and if S' is the inverse of S,
then

SU = SV

implies S'SU = S'SV,

or U = V.

Similarly US = VS

implies U = V.

Definition of a group.

Let S, T, U, ... be a set of definite operations, capable of being
carried out on a common object or set of objects, and let the set
contain--

(i.) the operation ST, S and T being any two operations of the set;

(ii.) the inverse operation of S, S being any operation of the set;
the set of operations is then called a group.

The number of operations in a group may be either finite or infinite.
When it is finite, the number is called the _order_ of the group, and
the group is spoken of as a _group of finite order_. If the number of
operations is infinite, there are three possible cases. When the group
is represented by a set of geometrical operations, for the
specification of an individual operation a number of measurements will
be necessary. In more analytical language, each operation will be
specified by the values of a set of parameters. If no one of these
parameters is capable of continuous variation, the group is called a
_discontinuous group_. If all the parameters are capable of continuous
variation, the group is called a _continuous group_. If some of the
parameters are capable of continuous variation and some are not, the
group is called a _mixed group_.

If S' is the inverse operation of S, a group which contains S must
contain SS', which produces no change on any possible object. This is
called the _identical operation_, and will always be represented by I.
Since S^pS^q = S^(p+q) when p and q are positive integers, and S^pS' =
S^(p-1) while no meaning at present has been attached to S^q when q is
negative, S' may be consistently represented by S^(-1). The set of
operations ..., S^(-2), S(-1), 1, S, S^2, ... obviously constitute a
group. Such a group is called a _cyclical_ group.

Subgroups, conjugate operations, isomorphism, &c.

It will be convenient, before giving some illustrations of the general
group idea, to add a number of further definitions and explanations
which apply to all groups alike. If from among the set of operations
S, T, U, ... which constitute a group G, a smaller set S', T', U', ...
can be chosen which themselves constitute a group H, the group H is
called a _subgroup_ of G. Thus, in particular, if S is an operation of
G, the cyclical group constituted by ..., S^(-2), S^(-1), 1, S, S^2,
... is a subgroup of G, except in the special case when it coincides
with G itself.

If S and T are any two operations of G, the two operations S and
T^(-1)ST are called _conjugate_ operations, and T^(-1)ST is spoken of
as the result of _transforming_ S by T. It is to be noted that since
ST = T^(-1), TS, T, ST and TS are always conjugate operations in any
group containing both S and T. If T transforms S into itself, that is,
if S = T^(-1)ST or TS = ST, S and T are called _permutable_
operations. A group whose operations are all permutable with each
other is called an _Abelian_ group. If S is transformed into itself by
every operation of G, or, in other words, if it is permutable with
every operation of G, it is called a _self-conjugate_ operation of G.

The conception of operations being conjugate to each other is extended
to subgroups. If S', T', U', ... are the operations of a subgroup H,
and if R is any operation of G, then the operations R^(-1)S'R,
R^(-1)T'R, R^(-1)U'R, ... belong to G, and constitute a subgroup of G.
For if S'T' = U', then R^(-1)S'R.R^(-1)T'R = R^(-1)S'T'R = R^(-1)U'R.
This subgroup may be identical with H. In particular, it is
necessarily the same as H if R belongs to H. If it is not identical
with H, it is said to be _conjugate_ to H; and it is in any case
represented by the symbol R^(-1)HR. If H = R^(-1)HR, the operation R
is said to be permutable with the subgroup H. (It is to be noticed
that this does not imply that R is permutable with each operation of
H.)

If H = R^(-1)HR, when for R is taken in turn each of the operations of
G, then H is called a _self-conjugate_ subgroup of G.

A group is spoken of as _simple_ when it has no self-conjugate
subgroup other than that constituted by the identical operation alone.
A group which has a self-conjugate subgroup is called _composite_.

Let G be a group constituted of the operations S, T, U, ..., and g a
second group constituted of s, t, u, ..., and suppose that to each
operation of G there corresponds a single operation of g in such a way
that if ST = U, then _st_ = u, where s, t, u are the operations
corresponding to S, T, U respectively. The groups are then said to be
_isomorphic_, and the correspondence between their operations is
spoken of as an _isomorphism_ between the groups. It is clear that
there may be two distinct cases of such isomorphism. To a single
operation of g there may correspond either a single operation of G or
more than one. In the first case the isomorphism is spoken of as
_simple_, in the second as _multiple_.

Two simply isomorphic groups considered abstractly--that is to say, in
regard only to the way in which their operations combine among
themselves, and apart from any concrete representation of the
operations--are clearly indistinguishable.

If G is multiply isomorphic with g, let A, B, C, ... be the operations
of G which correspond to the identical operation of g. Then to the
operations A^(-1) and AB of G there corresponds the identical
operation of g; so that A, B, C, ... constitute a subgroup H of G.
Moreover, if R is any operation of G, the identical operation of g
corresponds to every operation of R^(-1)HR, and therefore H is a
self-conjugate subgroup of G. Since S corresponds to s, and every
operation of H to the identical operation of g, therefore every
operation of the set SA, SB, SC, ..., which is represented by SH,
corresponds to s. Also these are the only operations that correspond
to s. The operations of G may therefore be divided into sets, no two
of which contain a common operation, such that the correspondence
between the operations of G and g connects each of the sets H, SH, TH,
UH, ... with the single operations 1, s, t, u, ... written below them.
The sets into which the operations of G are thus divided combine among
themselves by exactly the same laws as the operations of g. For if
st = u, then SH.TH = UH, in the sense that any operation of the set
SH followed by any operation of the set TH gives an operation of the
set UH.

The group g, abstractly considered, is therefore completely defined by
the division of the operations of G into sets in respect of the
self-conjugate subgroup H. From this point of view it is spoken of as
the _factor-group_ of G in respect of H, and is represented by the
symbol G/H. Any composite group in a similar way defines abstractly a
factor-group in respect of each of its self-conjugate subgroups.

It follows from the definition of a group that it must always be
possible to choose from its operations a set such that every operation
of the group can be obtained by combining the operations of the set
and their inverses. If the set is such that no one of the operations
belonging to it can be represented in terms of the others, it is
called a set of _independent generating_ operations. Such a set of
generating operations may be either finite or infinite in number. If
A, B, ..., E are the generating operations of a group, the group
generated by them is represented by the symbol {A, B, ..., E}. An
obvious extension of this symbol is used such that {A, H} represents
the group generated by combining an operation A with every operation
of a group H; {H1, H2} represents the group obtained by combining in
all possible ways the operations of the groups H1 and H2; and so on.
The independent generating operations of a group may be subject to
certain relations connecting them, but these must be such that it is
impossible by combining them to obtain a relation expressing one
operation in terms of the others. For instance, AB = BA is a relation
conditioning the group {A, B}; it does not, however, enable A to be
expressed in terms of B, so that A and B are independent generating
operations.

Transitivity and primitivity.

Let O, O', O", ... be a set of objects which are interchanged among
themselves by the operations of a group G, so that if S is any
operation of the group, and O any one of the objects, then O.S is an
object occurring in the set. If it is possible to find an operation S
of the group such that O.S is any assigned one of the set of objects,
the group is called _transitive_ in respect of this set of objects.
When this is not possible the group is called _intransitive_ in
respect of the set. If it is possible to find S so that any
arbitrarily chosen n objects of the set, O1, O2, ..., O_n are changed
by S into O'1, O'2, ..., O'n respectively, the latter being also
arbitrarily chosen, the group is said to be n-ply transitive.

If O, O', O", ... is a set of objects in respect of which a group G
is transitive, it may be possible to divide the set into a number of
subsets, no two of which contain a common object, such that every
operation of the group either interchanges the objects of a subset
among themselves, or changes them all into the objects of some other
subset. When this is the case the group is called _imprimitive_ in
respect of the set; otherwise the group is called _primitive_. A group
which is doubly-transitive, in respect of a set of objects, obviously
cannot be imprimitive.

Illustrations of the group idea.

The foregoing general definitions and explanations will now be
illustrated by a consideration of certain particular groups. To begin
with, as the operations involved are of the most familiar nature, the
group of rational arithmetic may be considered. The fundamental
operations of elementary arithmetic consist in the addition and
subtraction of integers, and multiplication and division by integers,
division by zero alone omitted. Multiplication by zero is not a
definite operation, and it must therefore be omitted in dealing with
those operations of elementary arithmetic which form a group. The
operation that results from carrying out additions, subtractions,
multiplications and divisions, of and by integers a finite number of
times, is represented by the relation x' = ax + b, where a and b are
rational numbers of which a is not zero, x is the object of the
operation, and x' is the result. The totality of operations of this
form obviously constitutes a group.

If S and T represent respectively the operations x' = ax + b and x' =
cx + d, then T^(-1)ST represents x' = ax + d - ad + bc. When a and b
are given rational numbers, c and d may be chosen in an infinite
number of ways as rational numbers, so that d - ad + bc shall be any
assigned rational number. Hence the operations given by x' = ax + b,
where a is an assigned rational number and b is any rational number,
are all conjugate; and no two such operations for which the a's are
different can be conjugate. If a is unity and b zero, S is the
identical operation which is necessarily self-conjugate. If a is unity
and b different from zero, the operation x' = x + b is an addition.
The totality of additions forms, therefore, a single conjugate set of
operations. Moreover, the totality of additions with the identical
operation, i.e. the totality of operations of the form x' = x + b,
where b may be any rational number or zero, obviously constitutes a
group. The operations of this group are interchanged among themselves
when transformed by any operation of the original group. It is
therefore a self-conjugate subgroup of the original group.

The totality of multiplications, with the identical operation, i.e.
all operations of the form x' = ax, where a is any rational number
other than zero, again obviously constitutes a group. This, however,
is not a self-conjugate subgroup of the original group. In fact, if
the operations x' = ax are all transformed by x' = cx + d, they give
rise to the set x' = ax + d(1 - a). When d is a given rational number,
the set constitutes a subgroup which is conjugate to the group of
multiplications. It is to be noticed that the operations of this
latter subgroup may be written in the form x' - d = a(x - d).

The totality of rational numbers, including zero, forms a set of
objects which are interchanged among themselves by all operations of
the group.

If x1 and x2 are any pair of distinct rational numbers, and y1 and y2
any other pair, there is just one operation of the group which changes
x1 and x2 into y1 and y2 respectively. For the equations y1 = ax1 + b,
y1 = ax2 + b determine a and b uniquely. The group is therefore doubly
transitive in respect of the set of rational numbers. If H is the
subgroup that leaves unchanged a given rational number x1, and S an
operation changing x1 into x2, then every operation of S^(-1)HS leaves
x2 unchanged. The subgroups, each of which leaves a single rational
number unchanged, therefore form a single conjugate set. The group of
multiplications leaves zero unchanged; and, as has been seen, this is
conjugate with the subgroup formed of all operations x' - d = a(x -
d), where d is a given rational number. This subgroup leaves d
unchanged.

The group of multiplications is clearly generated by the operations x'
= px, where for p negative unity and each prime is taken in turn.
Every addition is obtained on transforming x' = x + 1 by the different
operations of the group of multiplications. Hence x' = x + 1, and x' =
px, (p = -1, 3, 5, 7, ...), form a set of independent generating
operations of the group. It is a discontinuous group.

As a second example the group of motions in three-dimensional space
will be considered. The totality of motions, i.e. of space
displacements which leave the distance of every pair of points
unaltered, obviously constitutes a set of operations which satisfies
the group definition. From the elements of kinematics it is known that
every motion is either (i.) a translation which leaves no point
unaltered, but changes each of a set of parallel lines into itself; or
(ii.) a rotation which leaves every point of one line unaltered and
changes every other point and line; or (iii.) a twist which leaves no
point and only one line (its axis) unaltered, and may be regarded as a
translation along, combined with a rotation round, the axis. Let S be
any motion consisting of a translation l along and a rotation a round
a line AB, and let T be any other motion. There is some line CD into
which T changes AB; and therefore T^(-1)ST leaves CD unchanged.
Moreover, T^(-1)ST clearly effects the same translation along and
rotation round CD that S effects for AB. Two motions, therefore, are
conjugate if and only if the amplitudes of their translation and
rotation components are respectively equal. In particular, all
translations of equal amplitude are conjugate, as also are all
rotations of equal amplitude. Any two translations are permutable with
each other, and give when combined another translation. The totality
of translations constitutes, therefore, a subgroup of the general
group of motions; and this subgroup is a self-conjugate subgroup,
since a translation is always conjugate to a translation.

All the points of space constitute a set of objects which are
interchanged among themselves by all operations of the group of
motions. So also do all the lines of space and all the planes. In
respect of each of these sets the group is simply transitive. In fact,
there is an infinite number of motions which change a point A to A',
but no motion can change A and B to A' and B' respectively unless the
distance AB is equal to the distance A'B'.

The totality of motions which leave a point A unchanged forms a
subgroup. It is clearly constituted of all possible rotations about
all possible axes through A, and is known as the group of rotations
about a point. Every motion can be represented as a rotation about
some axis through A followed by a translation. Hence if G is the group
of motions and H the group of translations, G/H is simply isomorphic
with the group of rotations about a point.

The totality of the motions which bring a given solid to congruence
with itself again constitutes a subgroup of the group of motions. This
will in general be the trivial subgroup formed of the identical
operation above, but may in the case of a symmetrical body be more
extensive. For a sphere or a right circular cylinder the subgroups are
those that leave the centre and the axis respectively unaltered. For a
solid bounded by plane faces the subgroup is clearly one of finite
order. In particular, to each of the regular solids there corresponds
such a group. That for the tetrahedron has 12 for its order, for the
cube (or octahedron) 24, and for the icosahedron (or dodecahedron) 60.

The determination of a particular operation of the group of motions
involves six distinct measurements; namely, four to give the axis of
the twist, one for the magnitude of the translation along the axis,
and one for the magnitude of the rotation about it. Each of the six
quantities involved may have any value whatever, and the group of
motions is therefore a continuous group. On the other hand, a subgroup
of the group of motions which leaves a line or a plane unaltered is a
mixed group.

We shall now discuss (i.) continuous groups, (ii.) discontinuous groups whose order is not finite, and (iii.) groups of finite order. For proofs of the statements, and the general theorems, the reader is referred to the bibliography.

_Continuous Groups._

The determination of a particular operation of a given continuous group depends on assigning special values to each one of a set of parameters which are capable of continuous variation. The first distinction regards the number of these parameters. If this number is finite, the group is called a _finite_ continuous group; if infinite, it is called an _infinite_ continuous group. In the latter case arbitrary functions must appear in the equations defining the operations of the group when these are reduced to an analytical form. The theory of infinite continuous groups is not yet so completely developed as that of finite continuous groups. The latter theory will mainly occupy us here.

Sophus Lie, to whom the foundation and a great part of the development of the theory of continuous groups are due, undoubtedly approached the subject from a geometrical standpoint. His conception of an operation is to regard it as a geometrical transformation, by means of which each point of (n-dimensional) space is changed into some other definite point.

The representation of such a transformation in analytical form
involves a system of equations,

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

expressing x'1, x'2, ..., x'_n, the co-ordinates of the transformed
point in terms of x1, x2, ..., x_n, the co-ordinates of the original
point. In these equations the functions [f]_s are analytical functions
of their arguments. Within a properly limited region they must be
one-valued, and the equations must admit a unique solution with
respect to x1, x2, ..., x_n, since the operation would not otherwise
be a definite one.

From this point of view the operations of a continuous group, which
depends on a set of r parameters, will be defined 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),
(i.)

where a1, a2, ..., a_r represent the parameters. If this operation be
represented by A, and that in which b1, b2, ..., b_r are the
parameters by B, then the operation AB is represented by the
elimination (assumed to be possible) of x'1, x'2, ..., x'_n between
the equations (i.) and the equations

x"_s = [f]_s(x'1, x'2, ..., x'_n; b1, b2, ..., b_r),
(s = 1, 2, ..., n).

Since AB belongs to the group, the result of the elimination must be

x"_s = [f]_s(x1, x2, ..., x_n; c1, c2, ..., c_r),

where c1, c2, ..., c_r represent another definite set of values of the
parameters. Moreover, since A^(-1) belongs to the group, the result of
solving equations (i.) with respect to x1, x2, ..., x_n must be

x_s = [f]_s(x'1, x'2, ..., x'_n; d1, d2, ..., d_r),
(s = 1, 2, ..., n).

Conversely, if equations (i.) are such that these two conditions are
satisfied, they do in fact define a finite continuous group.

Infinitesimal operation of a continuous group.

It will be assumed that the r parameters which enter in equations (i.)
are independent, i.e. that it is impossible to choose r' (< r)
quantities in terms of which a1, a2, ..., a_r can be expressed. Where
this is the case the group will be spoken of as a "group of order r."
Lie uses the term "_r-gliedrige Gruppe_." It is to be noticed that the
word order is used in quite a different sense from that given to it in
connexion with groups of finite order.

In regard to equations (i.), which define the general operation of the
group, it is to be noticed that, since the group contains the
identical operation, these equations must for some definite set of
values of the parameters reduce to x'1 = x1, x'2 = x2, ..., x'_n =
x_n. This set of values may, without loss of generality, be assumed to
be simultaneous zero values. For if i1, i2, ..., i_r be the values of
the parameters which give the identical operation, and if we write

a_s = i_s + a, (s = 1, 2, ..., r),

then zero values of the new parameters a1, a2, ..., a_r give the
identical operation.

To infinitesimal values of the parameters, thus chosen, will
correspond operations which cause an infinitesimal change in each of
the variables. These are called infinitesimal operations. The most
general infinitesimal operation of the group is that given by the
system

[Pd][f]_s [Pd][f]_s [Pd][f]_s
x'_s - x_s = [delta]x_s = --------- [delta]a1 + --------- [delta]a2 + ... + --------- [delta]a_r, (s = 1, 2, ..., n),
[Pd]a1 [Pd]a2 [Pd]a_r

where, in [Pd][f]_s/[Pd]a_i, zero values of the parameters are to be
taken. Since a1, a2, ..., a_r are independent, the ratios of
[delta]a1, [delta]a2, ..., [delta]a_r are arbitrary. Hence the most
general infinitesimal operation of the group may be written in the
form

/ [Pd][f]_s [Pd][f]_s [Pd][f]_s\
[delta]x_s = ( e1--------- + e2--------- + ... + e_r--------- ) [delta]t, (s = 1, 2, ..., n),
\ [Pd]a1 [Pd]a2 [Pd]a_r /

where e1, e2, ..., e_r are arbitrary constants, and [delta]t is an
infinitesimal.

If F(x1, x2, ..., x_n) is any function of the variables, and if an
infinitesimal operation of the group be carried out on the variables
in F, the resulting increment of F will be

[Pd]F [Pd]F [Pd]F
------[delta]x1 + ------[delta]x2 + ... + -------[delta]x_n.
[Pd]x1 [Pd]x2 [Pd]x_n

If the differential operator

[Pd][f]1 [Pd] [Pd][f]2 [Pd] [Pd][f]_n [Pd]
-------- ------ + -------- ------ + ... + --------- -------
[Pd]a_i [Pd]x1 [Pd]a_i [Pd]x2 [Pd]a_i [Pd]x_n

be represented by X_i, (i = 1, 2, ..., r), then the increment of F is
given by

(e1X1 + e2X2 + ... + e_rX_r)F[delta]t.

When the equations (i.) defining the general operation of the group
are given, the coefficients [Pd][f]_s/[Pd]a_i, which enter in these
differential operators are functions of the variables which can be
directly calculated.

The differential operator e1X1 + e2X2 + ... + e_rX_r may then be
regarded as defining the most general infinitesimal operation of the
group. In fact, if it be for a moment represented by X, then (1 +
[delta]tX)F is the result of carrying out the infinitesimal operation
on F; and by putting x1, x2, ..., x_n in turn for F, the actual
infinitesimal operation is reproduced. By a very convenient, though
perhaps hardly justifiable, phraseology this differential operator is
itself spoken of as the general infinitesimal operation of the group.
The sense in which this phraseology is to be understood will be made
clear by the foregoing explanations.

We suppose now that the constants e1, e2, ..., e_r have assigned
values. Then the result of repeating the particular infinitesimal
operation e1X1 + e2X2 + ... + e_rX_r or X an infinite number of times
is some finite operation of the group. The effect of this finite
operation on F may be directly calculated. In fact, if [delta]t is the
infinitesimal already introduced, then

dF d^2F
-- = X.F, ---- = X.X.F, ...
dt dt^2

Hence

dF t^2 d^2F
F' = F + t-- + --- ---- + ...
dt 1.2 dt^2

t^2
= F + tX.F + --- X.X.F + ...
1.2

It must, of course, be understood that in this analytical
representation of the effect of the finite operation on F it is
implied that t is taken sufficiently small to ensure the convergence
of the (in general) infinite series.

When x1, x2, ... are written in turn for F, the system of equations

t^2
x'_s = (1 + tX + --- X.X + ...)x_s, (s = 1, 2, ..., n) (ii.)
1.2

represent the finite operation completely. If t is here regarded as a
parameter, this set of operations must in themselves constitute a
group, since they arise by the repetition of a single infinitesimal
operation. That this is really the case results immediately from
noticing that the result of eliminating F' between

t^2
F' = F + tX.F + --- X.X.F + ...
1.2

and

t'^2
F" = F' + t'X.F' + ---- X.X.F' + ...
1.2

is

(t + t')^2
F" = F + (t + t') X.F + ---------- X.X.F + ...
1.2

The group thus generated by the repetition of an infinitesimal
operation is called a _cyclical_ group; so that a continuous group
contains a cyclical subgroup corresponding to each of its
infinitesimal operations.

The system of equations (ii.) represents an operation of the group
whatever the constants e1, e2, ..., e_r may be. Hence if e1t, e2t,
..., e_rt be replaced by a1, a2, ..., a_r the equations (ii.)
represent a set of operations, depending on r parameters and belonging
to the group. They must therefore be a form of the general equations
for any operation of the group, and are equivalent to the equations
(i.). The determination of the finite equations of a cyclical group,
when the infinitesimal operation which generates it is given, will
always depend on the integration of a set of simultaneous ordinary
differential equations. As a very simple example we may consider the
case in which the infinitesimal operation is given by X =
x^2[Pd]/[Pd]x, so that there is only a single variable. The relation
between x' and t is given by dx'/dt = x'^2, with the condition that x'
= x when t = 0. This gives at once x' = x/(1 - tx), which might also
be obtained by the direct use of (ii.).

Relations between the infinitesimal operations of a finite continuous
group.

When the finite equations (i.) of a continuous group of order r are
known, it has now been seen that the differential operator which
defines the most general infinitesimal operation of the group can be
directly constructed, and that it contains r arbitrary constants. This
is equivalent to saying that the group contains r linearly independent
infinitesimal operations; and that the most general infinitesimal
operation is obtained by combining these linearly with constant
coefficients. Moreover, when any r independent infinitesimal
operations of the group are known, it has been seen how the general
finite operation of the group may be calculated. This obviously
suggests that it must be possible to define the group by means of its
infinitesimal operations alone; and it is clear that such a definition
would lend itself more readily to some applications (for instance, to
the theory of differential equations) than the definition by means of
the finite equations.

On the other hand, r arbitrarily given linear differential operators
will not, in general, give rise to a finite continuous group of order
r; and the question arises as to what conditions such a set of
operators must satisfy in order that they may, in fact, be the
independent infinitesimal operations of such a group.

If X, Y are two linear differential operators, XY - YX is also a
linear differential operator. It is called the "combinant" of X and Y
(Lie uses the expression _Klammerausdruck_) and is denoted by (XY). If
X, Y, Z are any three linear differential operators the identity
(known as Jacobi's)

(X(YZ)) + (Y(ZX)) + (Z(XY)) = 0

holds between them. Now it may be shown that any continuous group of
which X, Y are infinitesimal operations contains also (XY) among its
infinitesimal operations. Hence if r linearly independent operations
X1, X2, ..., X_r give rise to a finite continuous group of order r,
the combinant of each pair must be expressible linearly in terms of
the r operations themselves: that is, there must be a system of
relations

__k=r
(X_iX_j) = \ c_(ijk)X_k,
/__k=1

where the c's are constants. Moreover, from Jacobi's identity and the
identity (XY) + (YX) = 0 it follows that the c's are subject to the
relations

c_(ijt) + c_(jit) = 0, \
|
and >
|
[Sigma][s](c_(jks)c_(ist) + c_(kis)c_(jst) + c_(ijs)c_(kst)) = 0 /
(iii.)

for all values of i, j, k and t.

Determination of the distinct types of continuous groups of a given
order.

The fundamental theorem of the theory of finite continuous groups is
now that these conditions, which are necessary in order that X1, X2,
..., X_r may generate, as infinitesimal operations, a continuous group
of order r, are also sufficient.

For the proof of this fundamental theorem see Lie's works (cf.
Lie-Engel, i. chap. 9; iii. chap. 25).

If two continuous groups of order r are such that, for each, a set of
linearly independent infinitesimal operations X1, X2, ..., X_r and Y1,
Y2, ..., Y_r can be chosen, so that in the relations

(X_iX_j) = [Sigma]c_(ijs)X_s, (Y_iY_j) = [Sigma]d_(ijs)Y_s,

the constants c_(ijs) and d_(ijs) are the same for all values of i, j
and s, the two groups are simply isomorphic, X_s and Y_s being
corresponding infinitesimal operations.

Two continuous groups of order r, whose infinitesimal operations obey
the same system of equations (iii.), may be of very different _form_;
for instance, the number of variables for the one may be different
from that for the other. They are, however, said to be of the same
_type_, in the sense that the laws according to which their operations
combine are the same for both.

The problem of determining all distinct types of groups of order r is
then contained in the purely algebraical problem of finding all the
systems of r^3 quantities c_(ijs) which satisfy the relations

c_(ijt) + c_(ijt) = 0,

[Sigma] [c_(ijs)c_(skt) + c_(jks)c_(sit) + c_(kis)c_(sjt)] = 0.
s

for all values of i, j, k and t. To two distinct solutions of the
algebraical problem, however, two distinct types of group will not
necessarily correspond. In fact, X1, X2, ..., X_r may be replaced by
any r independent linear functions of themselves, and the c's will
then be transformed by a linear substitution containing r^2
independent parameters. This, however, does not alter the type of
group considered.

For a single parameter there is, of course, only one type of group,
which has been called cyclical.

For a group of order two there is a single relation

(X1X2) = [alpha]X1 + [beta]X2.

If [alpha] and [beta] are not both zero, let [alpha] be finite. The
relation may then be written ([alpha]X1 + [beta]X2, [alpha]^(-1)X2) =
[alpha]X1 + [beta]X2. Hence if [alpha]X1 + [beta]X2 = X'1, and
[alpha]^(-1)X2 = X'2, then (X'1X'2) = X'1. There are, therefore, just
two types of group of order two, the one given by the relation last
written, and the other by (X1X2) = 0.

Lie has determined all distinct types of continuous groups of orders
three or four; and all types of non-integrable groups (a term which
will be explained immediately) of orders five and six (cf. Lie-Engel,
iii. 713-744).

Self-conjugate subgroups. Integrable groups.

A problem of fundamental importance in connexion with any given
continuous group is the determination of the self-conjugate subgroups
which it contains. If X is an infinitesimal operation of a group, and
Y any other, the general form of the infinitesimal operations which
are conjugate to X is

t^2
X + t(XY) + --- ((XY)Y) + ....
1.2

Any subgroup which contains all the operations conjugate to X must
therefore contain all infinitesimal operations (XY), ((XY)Y), ...,
where for Y each infinitesimal operation of the group is taken in
turn. Hence if X'1, X'2, ..., X'_s are s linearly independent
operations of the group which generate a self-conjugate subgroup of
order s, then for _every_ infinitesimal operation Y of the group
relations of the form

__e=1
(X'_iY) = \ a_(ie)X'_e, (i = 1, 2, ..., s)
/__e=s

must be satisfied. Conversely, if such a set of relations is
satisfied, X'1, X'2, ..., X'_s generate a subgroup of order s, which
contains every operation conjugate to each of the infinitesimal
generating operations, and is therefore a self-conjugate subgroup.

A specially important self-conjugate subgroup is that generated by the
combinants of the r infinitesimal generating operations. That these
generate a self-conjugate subgroup follows from the relations (iii.).
In fact,

((X_iX_j)X_k) = [Sigma] c_(ijs)(X_sX_k).
s

Of the 1/2r(r - 1) combinants not more than r can be linearly
independent. When exactly r of them are linearly independent, the
self-conjugate group generated by them coincides with the original
group. If the number that are linearly independent is less than r, the
self-conjugate subgroup generated by them is actually a subgroup; i.e.
its order is less than that of the original group. This subgroup is
known as the derived group, and Lie has called a group _perfect_ when
it coincides with its derived group. A simple group, since it contains
no self-conjugate subgroup distinct from itself, is necessarily a
perfect group.

If G is a given continuous group, G1 the derived group of G, G2 that
of G1, and so on, the series of groups G, G1, G2, ... will terminate
either with the identical operation or with a perfect group; for the
order of G_(s+1) is less than that of G_s unless G_s is a perfect
group. When the series terminates with the identical operation, G is
said to be an _integrable_ group; in the contrary case G is called
_non-integrable_.

If G is an integrable group of order r, the infinitesimal operations
X1, X2, ..., X_r which generate the group may be chosen so that X1,
X2, ..., X_(r1), (r1 < r) generate the first derived group, X1, X2,
..., X_(r2), (r2 < r1) the second derived group, and so on. When they
are so chosen the constants c_(ijs) are clearly such that if r_p < i
<= r_(p+1), r_q < j <= r_(q+1), p >= q, then c_(ijs) vanishes unless
s <= r_(p+1).

In particular the generating operations may be chosen so that c_(ijs)
vanishes unless s is equal to or less than the smaller of the two
numbers i, j; and conversely, if the c's satisfy these relations, the
group is integrable.

Simple groups.

A simple group, as already defined, is one which has no self-conjugate
subgroup. It is a remarkable fact that the determination of all
distinct types of simple continuous groups has been made, for in the
case of discontinuous groups and groups of finite order this is far
from being the case. Lie has demonstrated the existence of four great
classes of simple groups:--

(i.) The groups simply isomorphic with the general projective group in
space of n dimensions. Such a group is defined analytically as the
totality of the transformations of the form

a_s, _1x1 + a_s, _2x2 + ... + a_s, _nx_n + a_(s, n + 1)
x'_s = --------------------------------------------------------, (s = 1, 2, ..., n),
a_(n+1), _1x1 + a_(n+1), _2x2 + ... + a_(n+1), _nx_n + 1

where the a's are parameters. The order of this group is clearly n(n +
2).

(ii.) The groups simply isomorphic with the totality of the projective
transformations which transform a non-special linear complex in space
of 2n - 1 dimensions with itself. The order of this group is n(2n +
1).

(iii.) and (iv.) The groups simply isomorphic with the totality of the
projective transformations which change a quadric of non-vanishing
discriminant into itself. These fall into two distinct classes of
types according as n is even or odd. In either case the order is
1/2n(n + 1). The case n = 3 forms an exception in which the
corresponding group is not simple. It is also to be noticed that a
cyclical group is a simple group, since it has no continuous
self-conjugate subgroup distinct from itself.

W. K. J. Killing and E. J. Cartan have separately proved that outside
these four great classes there exist only five distinct types of
simple groups, whose orders are 14, 52, 78, 133 and 248; thus
completing the enumeration of all possible types.

To prevent any misapprehension as to the bearing of these very general
results, it is well to point out explicitly that there are no
limitations on the parameters of a continuous group as it has been
defined above. They are to be regarded as taking in general complex
values. If in the finite equations of a continuous group the imaginary
symbol does not explicitly occur, the finite equations will usually
define a group (in the general sense of the original definition) when
both parameters and variables are limited to real values. Such a group
is, in a certain sense, a continuous group; and such groups have been
considered shortly by Lie (cf. Lie-Engel, iii. 360-392), who calls
them _real_ continuous groups. To these real continuous groups the
above statement as to the totality of simple groups does not apply;
and indeed, in all probability, the number of types of _real_ simple
continuous groups admits of no such complete enumeration. The effect
of limitation to real transformations may be illustrated by
considering the groups of projective transformations which change

x^2 + y^2 + z^2 - 1 = 0 and x^2 + y^2 - z^2 - 1 = 0

respectively into themselves. Since one of these quadrics is changed
into the other by the imaginary transformation

x' = x, y' = y, z' = z[root](-1),

the general continuous groups which transform the two quadrics
respectively into themselves are simply isomorphic. This is not,
however, the case for the _real_ continuous groups. In fact, the
second quadric has two real sets of generators; and therefore the real
group which transforms it into itself has two self-conjugate
subgroups, either of which leaves unchanged each of one set of
generators. The first quadric having imaginary generators, no such
self-conjugate subgroups can exist for the real group which transforms
it into itself; and this real group is in fact simple.

The adjunct group.

Among the groups isomorphic with a given continuous group there is one
of special importance which is known as the _adjunct_ group. This is a
homogeneous linear group in a number of variables equal to the order
of the group, whose infinitesimal operations are defined by the
relations

[Pd]
X_i=[Sigma] c_(ijs)x_i -------, (j = 1, 2, ..., r),
i, s [Pd]x_s

where c_(ijs) are the often-used constants, which give the combinants
of the infinitesimal operations in terms of the infinitesimal
operations themselves.

That the r infinitesimal operations thus defined actually generate a
group isomorphic with the given group is verified by forming their
combinants. It is thus found that (X_pX_q) = [Sigma][s]c_(pqs)X_s. The
X's, however, are not necessarily linearly independent. In fact, the
sufficient condition that [Sigma][j]a_jX_j should be identically zero
is that [Sigma][j]a_jc_(ijs) should vanish for all values of i and s.
Hence if the equations [Sigma][j]a_jc_(ijs) = 0 for all values of i
and s have r' linearly independent solutions, only r - r' of the X's
are linearly independent, and the isomorphism of the two groups is
multiple. If Y1, Y2, ..., Y_r are the infinitesimal operations of the
given group, the equations

[Sigma] a_jc_(ijs) = 0, (s, i = 1, 2, ..., r)
j

express the condition that the operations of the cyclical group
generated by [Sigma][j]a_jY_i should be permutable with every
operation of the group; in other words, that they should be
self-conjugate operations. In the case supposed, therefore, the given
group contains a subgroup of order r' each of whose operations is
self-conjugate. The adjunct group of a given group will therefore be
simply isomorphic with the group, unless the latter contains
self-conjugate operations; and when this is the case the order of the
adjunct will be less than that of the given group by the order of the
subgroup formed of the self-conjugate operations.

Continuous groups of the line of the plane, and of three-dimensional
space.

We have been thus far mainly concerned with the abstract theory of
continuous groups, in which no distinction is made between two simply
isomorphic groups. We proceed to discuss the classification and theory
of groups when their form is regarded as essential; and this is a
return to a more geometrical point of view.

It is natural to begin with the projective groups, which are the
simplest in form and at the same time are of supreme importance in
geometry. The general projective group of the straight line is the
group of order three given by

ax + b
x' = -------
cx + d'

where the parameters are the ratios of a, b, c, d. Since

x'3 - x'2 x' - x'1 x3 - x2 x - x1
--------- . -------- = ------- . ------
x'3 - x'1 x' - x'2 x3 - x1 x - x2

is an operation of the above form, the group is triply transitive.
Every subgroup of order two leaves one point unchanged, and all such
subgroups are conjugate. A cyclical subgroup leaves either two
distinct points or two coincident points unchanged. A subgroup which
either leaves two points unchanged or interchanges them is an example
of a "mixed" group.

The analysis of the general projective group must obviously increase
very rapidly in complexity, as the dimensions of the space to which it
applies increase. This analysis has been completely carried out for
the projective group of the plane, with the result of showing that
there are thirty distinct types of subgroup. Excluding the general
group itself, every one of these leaves either a point, a line, or a
conic section unaltered. For space of three dimensions Lie has also
carried out a similar investigation, but the results are extremely
complicated. One general result of great importance at which Lie
arrives in this connexion is that every projective group in space of
three dimensions, other than the general group, leaves either a point,
a curve, a surface or a linear complex unaltered.

Returning now to the case of a single variable, it can be shown that
any finite continuous group in one variable is either cyclical or of
order two or three, and that by a suitable transformation any such
group may be changed into a projective group.

The genesis of an infinite as distinguished from a finite continuous
group may be well illustrated by considering it in the case of a
single variable. The infinitesimal operations of the projective group
in one variable are d/dx, x(d/dx), x^2(d/dx). If these combined with
x^3(d/dx) be taken as infinitesimal operations from which to generate
a continuous group among the infinitesimal operations of the group,
there must occur the combinant of x^2(d/dx) and x^3(d/dx). This is
x^4(d/dx). The combinant of this and x^2(d/dx) is 2x^5(d/dx) and so
on. Hence x^_r(d/dx), where r is any positive integer, is an
infinitesimal operation of the group. The general infinitesimal
operation of the group is therefore [f](x)(d/dx), where [f](x) is an
arbitrary integral function of x.