In their report entitled "Training of Teachers of Elementary and Secondary Mathematics" a subcommittee of the International Commission on the Teaching of Mathematics named the following nine subjects as those which may later be expected to be involved in the preparation in pure mathematics of the prospective mathematics teacher: Calculus, differential equations, solid analytic geometry, projective geometry, theory of equations, theory of functions, theory of curves and surfaces, theory of numbers, and some group theory. The natural significance of the adjective some in connection with the last of these nine subjects would seem to be that the knowledge of the prospective teacher along the line of group theory need not be as thorough as that along the other lines named, but that he should know something about this subject.

It is evident that strictly speaking, the adjective some should have modified each of the other subjects named above, since no one can know all about any one of the broad fields covered by them, but the common usage of these terms seems to justify the conclusion that what is meant is that the prospective teacher should have a good course, or its equivalent, in each of these broad fields of pure mathematics, except possibly the last one. In regard to this field it might be sufficient if he had listened to a course of lectures devoted to it or had looked over some of the literature relating thereto.

Two questions naturally present themselves in this connection. The first of these is, Why should the teachers of elementary and secondary mathematics be expected to know anything about group theory? If good reasons exist for demanding such a knowledge, it is natural to ask, Why is a superficial knowledge in this field less objectionable than in the other fields noted above?

The former of these questions could scarcely embarrass anyone who possesses at least a slight knowledge of the modern literature on elementary and secondary mathematics. When one reads that such fundamental notions of geometry as point, line, and plane can be based upon group notions, and "that the idea of displacement, and consequently the idea of group, has played a preponderant part in the genesis of geometry" one can easily see why teachers of elementary geometry should know some group theory.

The teachers of elementary arithmetic and elementary algebra may have less conscious need of a knowledge of group theory than those of elementary geometry, but the former are likely to read about the groups and subgroups formed by certain sets of real or complex numbers with respect to multiplication, and the groups and subgroups formed by other sets of such numbers with respect to addition. Moreover, the group properties of then roots of unity may easily present themselves to such teachers as soon as they think just a little beyond what they are actually expected to teach.

Having established the fact that the teachers of elementary and secondary mathematics must know something about group theory in order to understand some of the best modern literature relating to the subjects which they are expected to teach, it remains to answer the second of the two questions raised above. Since there is an algebraic group theory, an analysis group theory and a geometry group theory, one might perhaps infer that the use of the adjective some in the list of subjęcts mentioned in the first paragraph of this article might have been due to a feeling that the subject of group theory was too broad to come within the mathematical purview of the prospective teacher, and that such a student should therefore confine his attention either to a special part of this subject or to a somewhat superficial survey of the whole field.

It is, however, less interesting to consider the possible reasons for the use of the adjective some in the given connection than to inquire what elements of group theory are of most importance to the prospective mathematics teacher. It might be supposed that these elements should include only one definition of the word group as a technical mathematical term. Since various writers use this term with different meanings, the reader who has only a single definition in mind is likely to be embarrassed there

1H. Poincare, The Monist, vol. 9 (1899), p. 32.

by and hence one definition of this term, may be worse than none. Perhaps the most important idea for the student to acquire early in regard to group theory is that there is not only one definition of the mathematical term group but there are several, and there is not only one group theory but there are several such theories.

How then can we know that a certain set of operators constitutes a group? With respect to a finite set there is little trouble, since all the definitions of finite discontinuous groups which are found in good modern literature are practically equivalent. This is, however, not the case as regards the definitions (explicit or implied) of an infinite group. In fact, such a standard work as the Encyklopädie der Mathematischen Wissenschaften contains on page 218 of Volume I a definition which is satisfied by the infinite set of numbers of the form.

when these numbers are combined by multiplication. It is evident that this set of operators does not involve the identity, since O is not a natural number, nor does it involve the inverse of any of its operators. If we would let q assume also the value 0, the set would include the identity but not the inverse of any of its other operators.

The most comprehensive definition of the technical term group is found on page 243 of tome I, volume 2, of the Encyclopédie des Sciences Mathématiques. It is as follows: "One says of an aggregate of objects that it forms a group in the most general sense of the word, when there is given a rule permitting to deduce from any two A and B of the objects of the aggregate a third object C of this same aggregate." It is evident that this definition of group does not imply the existence of an identity or of the inverse of an operator in the aggregate under consideration. In fact, the aggregate may contain various identities and various inverses of the same operator and still satisfy this definition of a group as results directly from the consideration of algebraic number fields, which are included under the term group in both the German and the French editions of the large mathematical encyclopedia.

One frequently finds in recent mathematical literature criticisms relating to the use of the term group with respect to a

A. Loewy, Archiv der Mathematik und Physik, vol. 9 (1905), p. 105.
'A similar definition appears in tome 3, volume 1, page 334, of this work.

set of operators which are said not to form a group by the authors of these criticisms. For instance, A. Loewy criticizes, in the article to which we referred, the definition of the term group as given in the large German mathematical encyclopedia, stating that it is necessary to add another condition in order to obtain a definition of an infinite group. On the other hand, the conditions imposed in the said encyclopedia definition are already much more restrictive than those imposed in the definition cited in the preceding paragraph and they include the latter. Hence it is clear that this particular criticism by an eminent authority is based on the assumption that such a definition as the one cited in the preceding paragraph is too general.

We refer to this matter in order to emphasize the fact that what appeared to many as an illegitimate use of the term group due to carelessness or ignorance has been legalized in recent years. The technical term group has thus been given a breadth of meaning comparable to that given to the term function in 1837 by a definition due to Dirichlet. This breadth of meaning is, however, not new but it embodies the fundamental notion which guided the earliest investigators in this field and to which certain explicit restrictions were added by later writers with a view to definite progress in certain directions.

Such restrictions remain desirable and will doubtless be continued, but instead of saying they are imposed on all groups we should say that they are imposed on the particular groups under consideration. Thus there arises the desirability of further classification of groups, and the study of groups in which the inverses of the operators are not present or in which a linear equation may have more than one solution. Up to the present it has been customary to study only those groups in which the inverse of each operator is present and in which a linear equation has only one solution. On the other hand, an equation of higher than the first degree can have more group elements for roots than the degree of this equation. These groups have been divided into four great classes named as follows: Finite, infinite discontinuous, finite continuous and infinite continuous.

The most important property of groups is the existence of subgroups, and hence subgroups should be considered immediately after some of the definitions of groups have been examined and illustrated. In particular, the facts that the order of a subgroup divides the order of the group whenever this order is finite, and that an invariant subgroup corresponds to the identity

of an isomorphic quotient group are of fundamental importance and should be included in the briefest course along this line. The subgroups of the continuous groups of movements of space corresponding to points and to lines respectively, are also of fundamental importance.

An important notion which is intimately connected with group theory but which is not confined to this subject is that of isomorphism. The special isomorphism in which there is a (1,1) correspondence between the elements of two groups is particularly important since it enables us to study once for all the fundamental properties of large categories of groups, and since it gave rise to the notion of abstract groups. Moreover, the possible isomorphisms of a group with itself are of fundamental importance in some of the applications of these groups. Hence the question of isomorphisms might well be considered immediately after that of subgroups, even if a thorough study of this question cannot be taken up until after a considerable body of theorems relating to group properties has been established.

The most important continuous group is composed of the transformations under which the properties of figures studied in elementary geometry are invariant. This group consists of movements, reflections, and similarity transformations and was called by F. Klein the principal or fundamental group. Two of its most important subgroups were noted above. It is interesting to note, in particular, that the dimensions of space are invariant under this group. As an instance of a group, in the most general sense, under whose transformations the dimensions of space are not invariant we may refer to the following:

which clearly transforms all points of the plane into those of a line passing through the origin, if we regard a and b as parameters and k as a constant.

The most important finite group is the cyclic group and there is one and only one such abstract group of each order. The cyclic group of order n is perhaps best known in the form of the n nth roots of unity. Among non-cyclic groups of finite order the dihedral groups, or groups of the regular polygons, are perhaps the most important. It must, however, be admitted that it is difficult to decide which are the most important among the hosts of finite and infinite groups intimately related with questions of elementary mathematics. Moreover, groups are social