Theory of Groups of Finite Order

The theory of groups of finite order may be said to date from the timeof Cauchy. To him are due the first attempts at classification with a view toforming a theory from a number of isolated facts. Galois introduced into thetheory the exceedingly important idea of a self-conjugate sub-group andthe corresponding division of groups into simple and composite. Moreoverby shewing that to every equation of finite degree there corresponds a groupof finite order on which all the properties of the equation depend Galoisindicated how far reaching the applications of the theory might be andthereby contributed greatly if indirectly to its subsequent developement.Many additions were made mainly by French mathematicians duringthe middle part of the century. The first connected exposition of the theorywas given in the third edition of M. Serrets Cours dAlgbre Suprieurewhich was published in 1866. This was followed in 1870 by M. Jordans Trait des substitutions et des quations algbriques. The greater part ofM. Jordans treatise is devoted to a developement of the ideas of Galoisand to their application to the theory of equations.No considerable progress in the theory as apart from its applicationswas made till the appearance in 1872 of Herr Sylows memoir Thormessur les groupes de substitutions in the fifth volume of the MathematischeAnnalen. Since the date of this memoir but more especially in recentyears the theory has advanced continuously 1882 appeared Herr Nettos Substitutionentheorie und ihre Anwen-dungen auf die Algebra in which as in M. Serrets and M. Jordans worksthe subject is treated entirely from the point of view of groups of substi-tutions. Last but not least among the works which give a detailed accountof the subject must be mentioned Herr Webers Lehrbuch der Algebra ofwhich the first volume appeared in 1895 and the second in 1896. In thelast section of the first volume some of the more important properties ofsubstitution groups are given. In the first section of the second volumehowever the subject is approached from a more general point of view anda theory of finite groups is developed which is quite independent of anyspecial mode of representing them.The present treatise is intended to introduce to the reader the main outlines of the theory of groups of finite order apart from any applications.The subject is one which has hitherto attracted but little attention in thiscountry; it will afford me much satisfaction if by means of this bookI shall succeed in arousing interest among English mathematicians in abranch of pure mathematics which becomes the more fascinating the moreit is studied.Cayleys dictum that a group is defined by means of the laws of combi-nation of its symbols would imply that in dealing purely with the theoryof groups no more concrete mode of representation should be used than isabsolutely necessary. It may then be asked why in a book which professesto leave all applications on one side a considerable space is devoted tosubstitution groups; while other particular modes of representation suchas groups of linear transformations are not even referred to. My answerto this question is that while in the present state of our knowledge manyresults in the pure theory are arrived at most readily by dealing with prop-erties of substitution groups it would be difficult to find a result thatcould be most directly obtained by the consideration of groups of lineartransformations.