Download e-book for iPad: A Course in the Theory of Groups (2nd Edition) (Graduate by Derek J. S. Robinson

By Derek J. S. Robinson

ISBN-10: 0387944613

ISBN-13: 9780387944616

"An first-class updated creation to the speculation of teams. it really is normal but accomplished, masking quite a few branches of workforce idea. The 15 chapters include the subsequent major themes: unfastened teams and shows, unfastened items, decompositions, Abelian teams, finite permutation teams, representations of teams, finite and limitless soluble teams, crew extensions, generalizations of nilpotent and soluble teams, finiteness properties." —-ACTA SCIENTIARUM MATHEMATICARUM

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Extra info for A Course in the Theory of Groups (2nd Edition) (Graduate Texts in Mathematics, Volume 80)

Example text

Each element of G has a unique expression of the form hn where h E Hand n E N. For example, the dihedral group D2n is a semidirect product of a cyclic group of order n and a group of order 2. ) Conjugation in N by an element h of H yields an automorphism h a of Nand 0:: h H ha is a homomorphism from H to Aut N. Observe that G is the direct product of Hand N if and only if 0: is the zero homomorphism. Conversely suppose that we are given two groups Hand N, together with a homomorphism 0:: H -+ Aut N.

Clearly D is a subgroup of C; in fact it is even a normal subgroup. In case A = {Al' A2 , ••• , An}, a finite set, we write D=G; "xG ; ' 2 x"·xG. ;'n Of course C = D in this case. Should the groups G;, be written additively, we shall speak of the direct sum of the G;" and write G;" $ G;'2 $ ... $ G;'n instead of G;" x G;'2 X ... x G;'n' For each A in A we define a function I;,: G;, -. C by agreeing that g1 shall be the vector whose A-component is g;, and whose other components are identity elements.

A p-subgroup of G which has this maximum order pa is called a Sylow psubgroup of G. We shall prove that Sylow p-subgroups of G always exist and that any two are conjugate-so, in particular, all Sylow p-subgroups of G are isomorphic. 16 (Sylow'S Theorem). Let G be a finite group and p a prime. Write IGI = pam where the integer m is not divisible by p. (i) Every p-subgroup of G is contained in a subgroup of order pa. In particular, since 1 is a p-subgroup, Sylow p-subgroups always exist. (ii) If np is the number of Sylow p-subgroups, np == 1 mod p.

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A Course in the Theory of Groups (2nd Edition) (Graduate Texts in Mathematics, Volume 80) by Derek J. S. Robinson

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