By Serge Lang (auth.)

This publication is meant as a easy textual content for a one-year path in Algebra on the graduate point, or as an invaluable reference for mathematicians and pros who use higher-level algebra. It effectively addresses the fundamental recommendations of algebra. For the revised 3rd version, the writer has additional workouts 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, maintaining classical themes yet introducing language and methods of pondering from class thought and homological algebra. It has affected all next graduate-level algebra books.
April 1999 Notices of the AMS, saying that the writer was presented the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.

The writer has a magnificent knack for providing the real and engaging rules of algebra in precisely the "right" method, and he by no means will get slowed down within the dry formalism which pervades a few components of algebra.
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We leave the first three statements to the reader, and prove the others . (iv) Let d l n . Let m = n/ d. Let f : Z � G be a surjective homomorphism . Then f(mZ) is a subgroup of G , and from the isomorphism Z/mZ = G /f( m Z) we conclude that f(mZ) has index m in G, whencef(mZ) has order d. Conversely , let H be a subgroup of order d. Then f - 1 (H) = mZ for some positive integer m, so H = f(mZ), Z/m Z = G/H, so n = md, m = n ld and H is uniquely determined . (v) Let A = (a) and B = (b) be cyclic groups of orders m , n , relatively prime .

Let S be one basis, with a finite number of elements m. Let T be another basis, and suppose that T has at least r elements. It will suffice to prove that r < m (one 42 G ROU PS I, §8 can then use symmetry). Let p be a prime number. Then BjpB is a direct sum of cyclic groups of order p, with m terms in the sum. Hence its order is pm. Using the basis T instead of S, we conclude that BjpB contains an r-fold product of cyclic groups of order p, whence pr < pm, and r < m, as was to be shown. ) The number of elements in a basis of a free abelian group A will be called the rank of A.

However, suppose n < m. Then the restriction of Em to Sn (viewed as a permutation of Jn leaving the elements of J not in Jn fixed) gives a homomorphism satisfying the conditions of Proposition 5 . 3 , so this restriction is equal to En . Thus Am n sn = A n . Next we prove some properties of the alternating group . m (a) A n is generated by the 3-cycles. Proof· Consider the product of two trans­ positions [ij] [rs] . If they have an element in common , the product is either the identity or a 3-cycle .

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