Primitive Groups, Vol. 1 (Classic Reprint)

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Excerpt from Primitive Groups, Vol. 1Some knowledge of Algebraic Numbers and of the ordinary Theory of Numbers is assumed to have been acquired by the reader by way of preparation for a serious study of the subject of which this volume treats.An apology may be in order for the arrangement of the subject matter. It was arranged as it is to meet the needs of actual instruction. The use of group characteristics, as developed by Frobenius, should be a familiar tool in the hands of the student as early as possible. There fore linear substitutions are taken up in the third chapter. From the point of view of strict logic this study of linear substitutions and of linear groups should be quite fully developed before those very special substitutions which we call permutations are considered. But the idea of groups of non-commutative Operations can, in the author's opinion, be best gained from a few lessons on the concrete and familiar permuta tions of a finite number of letters. Therefore the first two chapters are intended to familiarize the learner with the simpler processes used in Group Theory, to exhibit the fundamental theorems which admit of brie¿y worded proof, and to prepare the way for the more difficult developments of linear groups. Moreover, since any abstract group of finite order is isomorphic to some group of permutations, it would seem that sufficient generality can be attained if the phraseology of the abstract theory is ignored, as is done in this book.In talking of prime numbers it is admitted that it is a matter of in difference whether unity is included among the primes or not. May one be permitted the same license, if for the sake of convenience in stating certain theorems, the identical substitution alone is denied the dignity of being called a group The new terms Similar groups.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.