December 8, 1898] 



NATURE 



123 



algebra by a consecutive formulation of Kronecker's 

 abstractions without using the notion of a limit, and 

 restricts himself in the process to those parts of the group- 

 theory that are necessary for his purpose ; \'ogt provides 

 for French readers an equivalent of what Netto, in his 

 elder book, and Netto's translators, Battaglini and Cole, 

 had given to Germans, French, and English ; in Weber's 

 great treatise the theory appears as an important aspect, 

 but still only an aspect, of the problem of algebra. For 

 all these books the theory of groups is a means to an 

 end, or rather to one of multifarious ends which it can 

 be called upon to serve, viz. the algebraic solution of 

 equations. 



But in course of time the theory of groups has ever 

 more and more emancipated itself from algebra, whose 

 servant it originally was. Years ago Cayley threw out, 

 though he did not develop, the fundamental idea that 

 the notion of a group, in and for itself, is in no way 

 bound up with permutations and substitutions, but arises 

 whenever the effect of two operations performed succes- 

 sively upon an object is the same as that of a single 

 operation of the same kind. In the general theory of 

 groups thus founded, groups of finite order form a well- 

 marked division. To bring together the great mass of 

 single results yielded by exploring this division of the 

 theor)' from different sides, to take a comprehensive view 

 of them, and to exhibit them in a well-digested form, is 

 the problem that has been attempted for the first time in 

 the book before us, and it is solved in the happiest 

 manner. 



The author does not push abstraction to the point of 

 banishing from his book ail concrete methods of repre- 

 sentation of groups. He frequently uses properties of 

 groups of substitutions, in particular, not merely with a 

 view to making results plain to intuition, but also for the 

 deduction froii> them of properties of groups in the 

 abstract. He himself asks the question why other par- 

 ticular methods of representation of a group, e.g. by 

 means of homogeneous linear transformations, are not 

 employed in a similar way, and he answers it, as I think 

 rightly, in the words "that, while in the present state of 

 our knowledge, many results in the pure theory are 

 arrived at most readily by dealing with properties of 

 substitution groups, it would be difficult to find a result 

 that could be most directly obtained by the consideration 

 of groups of linear transformations." 



Believing that familiarity with symbolical calculations 

 concerning interchanges of letters is not to be assumed 

 on the part of his countrN-men, the author gives a sketch 

 of this theory in a short first chapter. The second 

 chapter begins with Cayleys abstract definition, cited 

 above, and carries the development of the general pro- 

 perties of a group as far as iJyck's theorem to the effect 

 that every group of finite order X can be represented as 

 a group of interchanges of N symbols. The third chapter 

 develops the notions : sub-group, self-conjugate sub- 

 group, simple group, isomorphism, factor groups. Then 

 follow in chapters four and five special investigations 

 relating to Abelian groups and to groups whose order 

 is a prime number ; in the latter the author has placed a 

 series of results of his own researches. The sixth chapter 

 brings us back with Sylow's theorem to the general 

 theory, and the pivot of the theory is found, in chapter 

 NO. 1 5 19, VOL. 59] 



seven, in the theorems on the composition-series of a 

 group. The three following chapters are especially con- 

 cerned with groups of substitutions, with the questions 

 of their transitivity and primitivity ; in addition to the 

 general theorems, they contain a great number of com- 

 pleted researches on special groups and types of groups ; 

 some of these are of intrinsic interest, and others serve 

 as vivid illustrations of the general theory. The eleventh 

 chapter treats of the isomorphism of a group with itself 

 on the lines followed by Holder and Frobenius. In the 

 twelfth and thirteenth chapters is explained the method 

 of representation of a group which was developed in a 

 general manner by Dyck, viz. the representation by 

 means of geometrical transformations of a surface 

 divided into regions, and especially by the transform- 

 ations effected by linear relations between complex 

 variables. .■Vn older method, due to Cayley, of represent- 

 ation by coloured diagrams leads in simple cases to 

 results easier to appreciate at a glance ; this method also 

 is here expounded and illustrated by a beautifully 

 executed coloured plate. The fourteenth chapter treats 

 of the representation of a group by means of systems of 

 linear congruences. There is a certain incongruity in 

 the fact that while this is inserted, the representation by 

 means of systems of linear equations is omitted ; but the 

 theory of this method would by itself furnish material 

 for a second book as large as the one before us. Perhaps 

 the author will some day give us such a book. On the 

 other hand we find here the extension of the theory 

 named to congruences holding among Galois' iniaginaries. 

 This was cultivated in his time by E. Mathieu, and has 

 been taken up recently by E. H. Moore and the author 

 of this treatise, in whose hands it has led, among other 

 remarkable results, to the knowledge of a new series of 

 types of finite groups. Finally the last chapter gives an 

 account of the most modem enumerations of types of 

 groups, in particular of simple groups of an order which 

 can be expressed in a prescribed manner as a product of 

 primes. 



The author has not attempted to give historical re- 

 ferences concerning the discovery of the older theorems : 

 but for the more modem literature, of about the last 

 twenty years, and right up to the time of publication, his 

 references are full and trustworthy. 



In respect of the completeness and exactness of its 

 matter, the work fills in a very acceptable manner a gap 

 in the literature of mathematics, and not merely of 

 mathematics in England. It brings to the special subject 

 a great wealth of material for the widening and deepen- 

 ing of knowledge ; while at the same time beginners, 

 under expert guidance, will be able to make a profitable 

 use of it. H. BtRKHARDT. 



OUR BOOK SHELF. 



Radiation : an Elementary Treatise on Electro-magnetic 



Radiation, and on Kontgen and Cathode Rays. By 



H. H. Francis Hyndman, B.Sc ;Lond.,. With a 



Preface by Prof. Silvanus P. Thompson, D.Sc, F.R.S. 



Pp. xviii -r 307. (London : Swan Sonnenschein and 



Co., Ltd. New York: The Macmillan Co., 1898.) 



The author considers chiefly those ponions of the 



subject which are somewhat neglected in most books, 



and in addition deals wiih the results of some of the 



