GROUPS,' 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 0, 0', 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 0 is represented by O.S, then O.S.S',O.S'.S., and O are the same object whatever object of the set 0 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 0 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.S2, O.S',... 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 0. UU' = O.S.T.T'.S' = O.SS' = 0, and O. U'U = O.T'.S'.S.T = O.T'T = 0, whatever object 0 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.
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 Definition the set contain— of a group.
(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,
'The word " group," which appears first in English in the sense of an assemblage of figures in an artistic design, picture, &c., is adapted from the Fr. groupe, which is to be referred to the Teutonic word meaning " knot," " mass," " bunch," represented in English by " crop " (q.v.). The technical mathematical sense is not older than 187o.
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 specication 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 SPS4=SPSs when p and q are positive integers, and SPS' =SP' while no meaning at present has been attached to S4 when q is negative, S' may be consistently represented by S '. The set of operations ..., S2, S1, i, S, S2, . . . obviously constitute a group. Such a group is called a cyclical group.
It will he convenient, before giving some illustrations of the
general group idea, to add a number of further definitions and exns which subgroups the planatioof operations apply T, U1.gro alike. If from .. hich constitute a group
conjugate G a smaller set S', T', U', ... can be chosen which themoperations, selves constitute a' group H, the group H is called a sub
ism, etc.  group Bc roup of G. Thus, in particular, if S is an operation of G,
the cyclical group constituted by . . . , i, S, S2, .. . 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'ST are called conjugate operations, and T'ST is spoken of as the result of transforming S by T. It is to be noted that since ST= T '. 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'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 selfconjugate 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'S'R, R'T'R, . . . belong to G, and constitute a subgroup of G. For if S'T'=U', then R'S'R.R'T'R=R'S'T'R= R'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'HR. If H=R'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'HR, when for R is taken in turn each of the operations of G, then H is called a selfconjugate subgroup of G.
A group is spoken of as simple when it has no selfconjugate subgroup other than that constituted by the identical operation alone. A group which has a selfconjugate 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 he 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 Ai 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'HR, and therefore H is a selfconjugate 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 t, 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 selfconjugate subgroup H. From this point of view it is spoken of as the factorgroup of G in respect of H, and is represented by the symbol G/H. Any composite group in a similar way defines abstractly a factorgroup in respect of each of its selfconjugate 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 calleda 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 Hi 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 IA, B} ; it does not, however, enable A to be expressed in terms of B, so that A and B are independent generating operations.
Let 0, 0', 0", . . 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 0 any one of the objects, then O. S 7, iis an object occurring in the set. If it is possible to find an tivity sad operation S of the group such that 0. S is any assigned one primiof the set of objects, the group is called transitive in respect tivity. 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, Ol, 02, ..., O„ are changed by S into O'1, 0'2, ..., O'„ respectively, the latter being also arbitrarily chosen, the group is said to be nply transitive.
If 0, 0', 0", . . . 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 doublytransitive, in respect of a set of objects, obviously cannot be imprimitive.
The foregoing general definitions and explanations will now be illustrated by a consideration of certain particular groups. To begin with, as the operation, involved are of the most familiar niustranature, thegroup of rational arithmetic maybe considered. boas of The fundamental operations of elementary arithmetic thegroup consist in the addition and subtraction of integers, and idea. 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'ST represents x'=ax+dad+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 dad+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 selfconjugate. 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 selfconjugate 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 selfconjugate 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(ta). 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(xd).
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 xl and x2 into y1 and y2 respectively. For the equations yl=axl+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 x,, and S an operation changing x, into x2, then every operation of SIHS 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(xd), 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+i by the different operations of the group of multiplications. Hence x'=x+I, and x'=px, (p= I, 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 threedimensional 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 T1ST leaves CD unchanged. Moreover, T1ST 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 selfconjugate 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) 6o.
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 r
x'.x,=Sxe=daeSal+ Sae+ . . . + Sat, (s=I, 2, . . . , n), 1 2
where, in afelaati, zero values of the parameters are to be taken. Since a1, a2, . , ar are independent, the ratios of Sal, Sa2, , Sa,. are arbitrary. Hence the most general infinitesimal operation of the group may be written in the form
ft.= . . . +e,aa') St, (s= I, 2, • , n),
110.1 a2 r
where e1, e2, ... , er are arbitrary constants, and & is an infinitesimal.
If F(x1, x2, ... , xn) 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
FSx1+Fx2 Sxz+ . . . +C)X,.
If the differential operator
afi a area afn a ~:+;ax2+...+n
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 (ndimensional) space is changed into some other definite point.
The representation of such a transformation in analytical form involves a system of equations,
x'e=fe(x1, x2r • xn) (s=I, 2, • •, n),
expressing x'1, x'2, ..., the coordinates of the transformed point
in terms of x,, x2, . xn, the coordinates of the original point. In these equations the functions fe are analytical functions of their arguments. Within a properly limited region they must be onevalued, and the equations must admit a unique solution with respect
to x1, x2, .. xn, 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'e=fe(x', x2, . . ., x,,; al, a2, ..., a,.), (s=I, 2, ..., n), (i.) where a1, a2, • • •r a,. represent the parameters. If this operation be represented by A, and that in which b,, b2, • • •, br 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"8=fe(x'1, x'2, . . .,x'n; b1, b2r • • , br), (s =I, 2r , n).
Since AB belongs to the group, the result of the elimination must be
x a=fa xl, x2r ,xn; c2,•, G.)r
where c1, c2, .. ., cr represent another definite set of values of the parameters. Moreover, since A1 belongs to the group, the result of solving equations (i.) with respect to x1, x2, .. ., x„ must be
xe=fe(x'I, x'2, ..., x',.; dl, d2, ..., (4), (s= I, 2r ..., n).
Conversely, if equations (i.) are such that these two conditions are satisfied, they do in fact define a finite 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' ( 

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