108 



SCIENCE. 



[N. S. Vol. IX. No. 212. 



matious of coordinates in geometry form a group; 

 so do the projections of a plane or of space; the 

 motions of space as a rigid body form the Eu- 

 clidean group of motions ; the n ! permutations 

 of n letters form a group ; the eight permuta- 

 tions of Ki, .T2, .T3, Xi which leave the function 

 Xi a,-2 + Xs Xi unchanged in form, form a group ; 

 the multiplication table, the operations of the 

 post office, the theory of the tides, psychological 

 phenomena, all embody characteristic groups. 

 A specially important class of groups, which 

 may serve to close the list, is that of the linear 

 transformations (which are formally identical 

 with geometric projections and with various 

 other operations). Thus the equation 



may be looked upon as defining an operation by 

 which any number z is connected into a corre- 

 sponding number z'. If we have two of these 

 operations, and if, having applied the one to z, 

 getting z' as a result, we apply the other to z', 

 getting z'^ as a result, then an examination will 

 show that z" is itself a linear function of 2, i. e. , 

 the product of two linear transformations is a 

 linear transformation. 



Prepare now for a step into the abstract. In 

 expressing ourselves in terms of 'operations' 

 we have been walking on the crutches of the 

 concrete. But if we designate the operations 

 of a group hy A, B, C, . . . , their products ^4 B, 

 BC, . . . have a definite mode of formation, 

 constituting an algebra, and we will now throw 

 away the ' operations ' and keep the symbols 

 and their algebra. The symbols are now ' ele- 

 ments,' and if these elements form a group 

 the product ^i? is identified by the algebra with 

 some element C of the same group. Two other 

 properties have to be added to make the defini- 

 tion of a group precise : (1) the algebra must be 

 associative, i. e., (AB).C= A.{BC), and (2) if 

 AB = AC then B=C and if AB = CB then A 

 = G. Algebras can, of course, be constructed 

 which omit these conditions, but they are not 

 algebras of groups. 



The order of a group is the number of its ele- 

 ments. A group may be of finite or Infinite 

 order, e. g., all the rotations of a sphere about 

 its diameter form a,u infinite group ; those of 



them which turn into itself a regular polyhedron 

 inscribed in the sphere form a finite group. In- 

 finite groups are only touched on in Burnside's 

 book. Access to their theory is most readily 

 had through Lie's works. Burnside's opening 

 chapter on abstract groups (ChaiJter 2) is not so 

 happily executed as Weber's treatment (Vol. II., 

 Chapter 1), which is a masterpiece {Cf. also 

 Frobenius's ' Ueber endlicheGruppen,' Berliner 

 Sitzungsberichte, 1895, p. 163). Buruside re- 

 tains the operations and makes use of their con- 

 crete qualities in discussing properties which 

 are better treated in the pure abstract. 



From the mere definition of a group it is pos- 

 sible to raise a considerable crop of properties 

 without any artificial fertilizer. Add the ideas 

 of isomorphism and transformation, and con- 

 sider the groups whose elements are commuta- 

 tive (Chapter 3), and those whose orders are 

 powers of single prime numbers (Chapter 4), 

 and the wilderness fairly blooms. Even the 

 non -specialist may rapidly make his way 

 through the easy roads and add valuable ideas 

 to his stock as he goes. He can hardly do bet- 

 ter than to read this book, which gives a very 

 clear and straightforward treatment of these 

 general matters. But this is mere surface pro- 

 duction. Underneath is gold, but only the 

 Frobenius brand of dynamite will reach that. 

 More than twenty-five years ago a solitary pros- 

 pector, Sylow, found the lode and worked it 

 with good results as far as he could follow it. 

 Others have tried new leads, but none have 

 accomplished anything remarkable until the 

 work of Frobenius, who in the past ten years 

 or so, and more particularly in his articles pub- 

 lished in the Berliner Sitzungsberichte for 1895-6 

 has opened up a vast wealth of new relations, 

 at the same time revising and enriching the ear- 

 lier methods, nomenclature, and general point of 

 view. Some of the most prominent of Frobeni- 

 us's results are discussed in Chapter 6. Another 

 line of ideas, which, however, dates back in its 

 beginning as far as Galois, and has been im- 

 proved especially by Holder, the theory of com- 

 position of a group, is discussed in Chapter 7. 

 The three following chapters are devoted to an 

 extensive discussion of substitution groups, 

 whose theory has also been considerably ex- 

 tended of recent j'ears. The theory of isomor- 



