By Serge Lang (auth.)

This ebook is meant as a simple textual content for a one-year direction in Algebra on the graduate point, or as an invaluable reference for mathematicians and pros who use higher-level algebra. It effectively addresses the elemental suggestions of algebra. For the revised 3rd version, the writer has additional routines and made various corrections to the text.

Comments on Serge Lang's Algebra:

Lang's Algebra replaced the way in which graduate algebra is taught, holding classical themes yet introducing language and methods of considering from classification idea and homological algebra. It has affected all next graduate-level algebra books.*April 1999 Notices of the AMS, saying that the writer **was offered the Leroy P. Steele Prize for Mathematical **Exposition for his many arithmetic books.*

The writer has a magnificent knack for featuring the real and fascinating rules of algebra in precisely 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 assessment of the 1st edition*

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**Extra info for Algebra**

**Example text**

Let G be a group, and let G = G t ::J G 2 ::J · · · ::J G r = {e}, G = H t ::J H 2 ::J · · · ::J Hs = {e} be normal towers of subgroups, ending with the trivial group. We shall say that these towers are equivalent if r = s and if there exists a permutation of the 22 I , §3 G ROU PS indices i = 1 , . . , r - 1, written i t-+ i ', such that � Gi/Gi + 1 H i · /H i ' + 1 · In other words, the sequences of factor groups in our two towers are the same, up to isomorphisms, and a permutation of the indices.

Thus when G is a finite group, the above formula reads {G : 1) = L { G : G x) xeC where C is a set of representatives for the distinct conjugacy classes, and the sum is taken over all x E C. This formula is also called t h e class formula. 30 G ROUPS I, §5 The class formula and the orbit decomposition formula will be used systematically in the next section on Sylow groups , which may be viewed as providing examples for these formulas . Readers interested in Sylow groups mayjump immediately to the next section .

L For each fixed point s; we have Hs; = H . For s; not fixed , the index (H : Hs) is divisible by p, so (a) follows at once . Parts (b) and (c) are special cases of (a) , thus proving the lemma . Remark. fixed points . In Lemma 6 . 3 (c) , if H has one fixed point, then H has at least p Theorem 6. 4. finite group. (i) If H is a p-subgroup of G, then H is contained in some p-Sylow subgroup. I , §6 SYLOW SUBG ROU PS (ii) All p-Sylow subgroups are conjugate. (iii) The number of p-Sylow subgroups of G is = 35 1 mod p.