Group (mathematics)

“ A group is a class of relations possessing a special property. Thus the modern ideas, which have so powerfully extended and unified the subject, have loosened its connexion with “number” and “quantity,” while bringing ideas of form and structure into increasing prominence. Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics. All the world, including savages who cannot count beyond five, daily “apply” theorems of number. ”

“ You cannot really create an idea: it happens—and you have to be on the alert to seize it when it does so happen.
It will be seen from the illustration that there are ten of these checks on a ring, numbered 1 to 9 and 0. The puzzle is to divide them into three groups without taking any off the ring, so that the first group multiplied by the second makes the third group. For example, we can divide them into the three groups, 2—8 9 7—1 5 4 6 3, by bringing the 6 and the 3 round to the 4, but unfortunately the first two when multiplied together do not make the third. ”

“ Although the notion of group is one of the most fundamental ones in mathematics, yet it is one which is more useful to arrive at reasons for certain results and at connections between apparently widely separated developments than to furnish methods for attaining these results or developments. Its greatest service so far has been its unifying influence and its usefulness in proving the possibility or the impossibility of certain operations. ”

“ In our century the conceptions substitution and substitution group, transformation and transformation group, operation and operation group, invariant, differential invariant and differential parameter, appear more and more clearly as the most important conceptions of mathematics.—Lie, Sophus. ”

“ The certainty of mathematics depends upon its complete abstract generality. But we can have no à priori certainty that we are right in believing that the observed entities in the concrete universe form a particular instance of what falls under our general reasoning. To take another example from arithmetic. It is a general abstract truth of pure mathematics that any group of forty entities can be subdivided into two groups of twenty entities. ”

“ 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. ”

“ 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. Illustrations of the group idea. 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. ”

“ The totality of the rational numbers clearly becomes a group when they are combined with respect to either of the operations addition or multiplication. The same remark applies evidently to all the real numbers as well as to all the complex numbers. These additive groups of infinite order are frequently represented by the equation , where may assume all the values of one of the given groups. If a may assume all real values the group is said to be continuous. ”

“ Yet I suppose that no mathematician would be disposed, at the present time, to accept any brief definition of the degree of closeness or remoteness of relation to ordinary experience which shall serve to distinguish a trivial from a genuinely significant branch of mathematical theory. In general, a mathematician who is devoted to the theory of functions, or to group theory, appears to spend little time in attempting to show why the development of the consequences of his postulates is a significant enterprise. ”

“ 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. ”

“ The aggregate of all individual things occurring in a definite concept, or the common characteristics of which make up this concept, is called a group. Such a group may consist of a limited or finite number of members, or may be unlimited, according to the nature of the concepts that characterize it. Thus, all the integers form an unlimited or infinite group, while the integers between ten and one hundred (or the two-digit numbers) form a limited or finite group. ”

“ It is to be noted that the concept of quantity has not yet been considered; what we have gained is the concept of number. The single things or members in this number are quite arbitrary, and especially they do not need to be alike in any manner. Every number forms a group-type, and arithmetic or the science of numbers has the task of investigating the properties of these different types with reference to their division and combination. If this is done in general form, without attention to the special amount of the number, the corresponding science is called algebra. ”

“ The modern mathematical conceptions of what is called group theory have already received very wide and significant applications, and promise to bring into unity regions of research which, until recently, appeared to have little or nothing to do with one another. Quite lately, however, there are signs that group theory will soon prove to be of importance for the definition of some of the fundamental concepts of that most refractory branch of philosophical inquiry, æsthetics. ”

“ The main principle of division must of course be natural affinity; the classes formed must be natural groups: and the formation of these has already been sufficiently treated of. But the principles of natural grouping must be applied in subordination to the principle of a natural series. The groups must not be so constituted as to place in the same group things which ought to occupy different points of the general scale. ”

“ 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. ”

“ Another large class of discontinuous groups, which have far-reaching applications in analysis, are those which arise in the first instance from purely geometrical considerations. By the combination and repetition of a finite number of geometrical Discontinuous groups arising from geometrical operations. operations such as displacements, projective transformations, inversions, &c., a discontinuous group of such operations will arise. ”

“ But the complex obtained in this manner is always accidental with reference to the number and the richness of its members. A regular and complete structure of all possible groups is evidently obtained only if we start from a group of one member or from a simple thing, and by the addition of one member at a time make further groups out of those that we have. ”

“ In general, if the equation is given arbitrarily, the group will be the symmetric group. The necessary and sufficient condition that the equation may be soluble by radicals is that its group should be a soluble group. When the coefficients in an equation are rational integers, the determination of its group may be made by a finite number of processes each of which involves only rational arithmetical operations. These processes consist in forming resolvents of the equation corresponding to each distinct type of subgroup of the symmetric group whose degree is that of the equation. ”

“ Number, space, continuity, functional correspondence or dependence, group-structure—these are examples of such significant ideas; the postulates or ideal assumptions upon which the theory of such ideas depends are significant postulates, and are not the mere conventions of an arbitrary game. But now what constitutes the significance of an idea, or of an abstract mathematical theory? What gives an idea a worthy place in the whole scheme of human ideas? Is it the possibility of finding a physical application for a mathematical theory which for us decides what is the value of the theory? ”

“ It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. ”

“ If is any number, the relation of to is one-one; so is the relation of to or to . When we are considering only positive numbers, the relation of to is one-one; but when negative numbers are admitted, it becomes two-one, since and have the same square. These instances should suffice to make clear the notions of one-one, one-many, and many-one relations, which play a great part in the principles of mathematics, not only in relation to the definition of numbers, but in many other connections. ”

“ Moreover, the determinant of the c’s is the product of the determinant of the a’s and the determinant of the b’s. Hence equations (ii.) , where the parameters are restricted to be integers of a given corpus, define a discontinuous group; and if the determinant of the coefficients is limited to the value unity, they define a discontinuous group which is a (self-conjugate) subgroup of the previous one.
The simplest case which thus presents itself is that in which there are two variables while the coefficients are rational integers. ”

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

“ There will be no royal road to the discovery of the best system,690 and it will even be impossible to lay down rules of procedure to assist those who are in search of a good arrangement. The only logical rule would be as follows:—Having given certain objects, group them in every way in which they can be grouped, and then observe in which method of grouping the correlation of properties is most conspicuously manifested. ”

“ The ordinal number ȯ is the class whose sole member is the null relation—that is, the relation which never holds between any pair of entities. The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals. Here also it can be seen that the science of the finite ordinals is a particular subdivision of the general theory of classes and relations. Thus the illusory nature of the traditional definition of mathematics is again illustrated. ”

“ The group of 1,000 words from 'Vanity Fair' enumerated above is thus graphically represented by the continuous line in Fig. 1, and the method of constructing the characteristic curve will be readily understood by comparing this with the numbers given. As a thousand is a very small number in a problem of this kind, the curve representing any single group of that number of words is practically certain to differ more or less from that of any other such group. In Fig. 1 the dotted line represents a group of 1,000 Fig. 1. Two Groups—1000 each—Vanity Fair. ”

“ On the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. The three, for instance, are regarded as a whole when we name them three. If, however, we count these three as one, two, three, then the number of times we count is an abstract number. Thus number in the abstract is the number of times that the act of counting is performed in any particular case. ”

“ We have thus far viewed mathematical science only as a whole, without paying any regard to its divisions. We must now, in order to complete this general view, and to form a just idea of the philosophical character of the science, consider its fundamental division. ”

“ That such an influence upon the judgment of number should have been exercised by the factors considered seems in many cases to receive an adequate account on the principle of association. Our practical experience in the simultaneous variability of number and certain other characteristics of a group of objects has been such as to lead us into illusions when the two no longer vary together. In such a case, when we have no time to count, we are actually led to see a group as smaller or larger in accordance with the variations perceived in the associated factor. ”