By Serge Lang

ISBN-10: 146130041X

ISBN-13: 9781461300410

This booklet is meant as a simple textual content for a one-year path in Algebra on the graduate point, or as an invaluable reference for mathematicians and execs who use higher-level algebra. It effectively addresses the elemental ideas of algebra. For the revised 3rd variation, the writer has further routines and made a number of corrections to the text.

Comments on Serge Lang's Algebra:
"Lang's Algebra replaced the best way graduate algebra is taught, conserving classical subject matters yet introducing language and methods of considering from type thought and homological algebra. It has affected all next graduate-level algebra books."
-April 1999 Notices of the AMS, saying that the writer used to be presented the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.

"The writer has a magnificent knack for proposing the $64000 and engaging rules of algebra in exactly the "right" manner, and he by no means will get slowed down within the dry formalism which pervades a few elements of algebra."
-MathSciNet's evaluation of the 1st variation

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Sample text

I One could also describe this isomorphism using (i) and (ii). We leave it to the reader to show that we have a commutative diagram o ---+ H ---+ G ---+ j~" 1 can GjH - - - + 0 l;, 0---+ HjK - - - + GjK - - - + GjH - - - + 0 where the rows are exact. (iv) Let G be a group and let H, K be two subgroups. Assume that H is contained in the normalizer of K . Then H n K is obviously a normal subgroup of H, and equally obviously H K = KH is a subgroup of G. There is a surjective homomorphism H -+ HK jK associating with each x E H the coset xK of K in the group H K.

Conversely , it is clear that mnZ is contained in the kernel , so the kernel is mnZ . The ima ge of Z ~ A x B is surj ective by the Chi nese remainder theorem . Thi s prov es (v) . 2 the re are at least two factors in the direct product (or sum) decomposition , and each contains a cycl ic subgroup of order p , whence G contains their direct product (or sum) . Statement (vi) is, of course, easier to prove than the full structure theorem , and it is a good exercise for the reader to formulate the simpler arguments which yield (vi) directly.

From such an operation we obtain a mapping G x s-« S, 26 GROUPS I, §5 which to each pair (x, s) with x E G and s E S associates the element 7Tx (S) . We often abbreviate the notation and write simply xs instead of 7TxCs) . r, y E G and s E S, we have x(ys) = (xy)s. If e is the unit eLement of G, then es = s for all s E S. Conversely, if we are given a mapping G x S ~ S, denoted by (x, s) M xs, satisfying these two properties, then for each x E G the map s t-+ xs is a permutation of S, which we then denote by nx(s).

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Algebra by Serge Lang

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