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).