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 SpSq = Sp+q when p and q are positive integers, and SpS′ = Sp−1 while no meaning at present has been attached to Sq when q is negative, S′ may be consistently represented by S−1. The set of operations . . ., S−2, S−1, 1, S, S2, . . . obviously constitute a group. Such a group is called a cyclical group.
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 themselvesSubgroups, conjugate operations, isomorphism, &c. 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, 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−1ST are called conjugate operations, and T−1ST 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−1ST 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−1S′R, R−1T′R, R−1U′R, . . . belong to G, and constitute a subgroup of G. For if S′T′ = U′, then R−1S′R·R−1T′R = R−1S′T′R = R−1U′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−1HR. If H = R−1HR, 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−1HR, 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−1HR, 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.
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 Transitivity and primitivity.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, . . ., On 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.
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 arithmeticIllustrations of
the group idea.
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−1ST 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.
