Theory of Groups of Finite Order

Theory of Groups of Finite Order

The theory of groups of finite order may be said to date from the time
of Cauchy. To him are due the first attempts at classification with a view to
forming a theory from a number of isolated facts. Galois introduced into the
theory the exceedingly important idea of a self-conjugate sub-group, and
the corresponding division of groups into simple and composite. Moreover,
by shewing that to every equation of finite degree there corresponds a group
of finite order on which all the properties of the equation depend, Galois
indicated how far reaching the applications of the theory might be, and
thereby contributed greatly, if indirectly, to its subsequent developement.
Many additions were made, mainly by French mathematicians, during
the middle part of the century. The first connected exposition of the theory
was given in the third edition of M. Serretâ€™s â€œ Cours dâ€™AlgÃ¨bre SupÃ©rieure,â€
which was published in 1866. This was followed in 1870 by M. Jordanâ€™s
â€œ TraitÃ© des substitutions et des Ã©quations algÃ©briques.â€ The greater part of
M. Jordanâ€™s treatise is devoted to a developement of the ideas of Galois
and to their application to the theory of equations.
No considerable progress in the theory, as apart from its applications,
was made till the appearance in 1872 of Herr Sylowâ€™s memoir â€œ ThÃ©orÃ¨mes
sur les groupes de substitutions â€ in the fifth volume of the Mathematische
Annalen. Since the date of this memoir, but more especially in recent
years, the theory has advanced continuously.
In 1882 appeared Herr Nettoâ€™s â€œ Substitutionentheorie und ihre Anwen-dungen auf die Algebra,â€ in which, as in M. Serretâ€™s and M. Jordanâ€™s works,
the 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 account
of the subject must be mentioned Herr Weberâ€™s â€œ Lehrbuch der Algebra,â€ of
which the first volume appeared in 1895 and the second in 1896. In the
last section of the first volume some of the more important properties of
substitution groups are given. In the first section of the second volume,
however, the subject is approached from a more general point of view, and
a theory of finite groups is developed which is quite independent of any
special 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 this
country; it will afford me much satisfaction if, by means of this book,
I shall succeed in arousing interest among English mathematicians in a
branch of pure mathematics which becomes the more fascinating the more
it is studied.
Cayleyâ€™s dictum that â€œa group is defined by means of the laws of combi-nation of its symbolsâ€ would imply that, in dealing purely with the theory
of groups, no more concrete mode of representation should be used than is
absolutely necessary. It may then be asked why, in a book which professes
to leave all applications on one side, a considerable space is devoted to
substitution groups; while other particular modes of representation, such
as groups of linear transformations, are not even referred to. My answer
to this question is that while, in the present state of our knowledge, many
results 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 that
could be most directly obtained by the consideration of groups of linear
transformations.
Country USA
Binding Kindle Edition
ReleaseDate 2013-02-26
Format Kindle eBook

Country	USA
Binding	Kindle Edition
ReleaseDate	2013-02-26
Format	Kindle eBook