Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1897 edition. Excerpt: ... Corollary. If H is Abelian, no two operations of H which are not conjugate in / can be conjugate in G. Hence the number of distinct sets of conjugate operations in G, which have powers of p for their orders, is the same as the number of such sets in /. 82. Suppose next that Q is not self-conjugate in H. Then every operation that transforms Q into P must transform H into a sub-group of order pa in which P is not self-conjugate. Of the sub-groups of order pa, to which P belongs and in which P is not self-conjugate, choose H' so that, in H', P forms one of as small a number of conjugate operations or sub-groups as possible. Let g be the greatest sub-group of H' that contains P self-conjugately. Among the sub-groups of order pa that contain P self-conjugately, there must be one or more to which g belongs. Let H be one of these; and suppose that h and h! are the greatest sub-groups of H and H' respectively that contain g self-conjugately. The orders of both h and h' must (Theorem II, § 55) be greater than the order of g; and in consequence of the assumption made with respect to H every sub-group, having a power of p for its order and containing h, must contain P self-conjugately. Now consider the sub-group h, h'. Since it does not contain P self-conjugately, its order cannot be a power of p. Also if pP is the highest power of p that divides its order, it must contain more than one sub-group of order pP. For any sub-group of order pP, to which h belongs, contains P selfconjugately; and any sub-group of order pP, to which h' belongs, does not. Suppose now that S is an operation of h, h', having its order prime to p and transforming a sub-group of h, h' of order pP, to which h belongs, into one to which h' belongs. Then S cannot be...
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Book Description
William Burnside (1852–1927) was one of the founders of the modern theory of finite groups. This book, originally published in 1897, is reissued here in the updated second edition of 1911. It includes an account of the character theory of Ferdinand Georg Frobenius (1849–1917), which influenced Burnside's research.
About the Author
British mathematician William Burnside (1852–1927) studied at Cambridge University and was later appointed Professor of Mathematics at the Royal Naval College in Greenwich, UK, where he spent his career. Burnside published more than 150 papers on mathematical topics. Cambridge University Press published the first edition of his classic book, Theory of Groups of Finite Order, in 1897 and in an expanded second edition in 1911. For the next 40-plus years, this book was the classic introduction to group theory, a worthy ancestor of Marshall Hall's The Theory of Groups, first published in 1959 and also now reprinted by Dover.
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Theory of Groups of Finite Order
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