By Serge Lang

This publication 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 pros who use higher-level algebra. It effectively addresses the elemental techniques of algebra. For the revised 3rd variation, the writer has additional workouts and made a variety of corrections to the text.

Comments on Serge Lang's Algebra:

"Lang's Algebra replaced the best way graduate algebra is taught, conserving classical issues yet introducing language and methods of considering from class idea 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 offered the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.

"The writer has a powerful knack for featuring 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 components of algebra."

-MathSciNet's overview of the 1st version

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**Example text**

G/H ~ 1m!.!... G'. Here, j is the inclusion of Im j" in G'. (ii) Let G be a group and H a subgroup. Let N be the intersection of all normal subgroups containing H. Then N is normal, and hence is the smallest normal subgroup of G containing H. Letj" : G -+ G' be a homomorphism whose kernel contains H. Then the kernel of! contains N, and there exists a unique homomorphism j, : G/N -+ G', said to be induced by f, making the following diagram commutative: G~G' \ J. GIN As before,

Furthermore , we have G = AN = NA , and N n A = {id}. In the terminology of Exercise 12, G is the semidirect product of A and N . Ta,b(X) * Let H be a subgroup of G. Then H is obviously a normal subgroup of its normalizer N H' We leave the following statements as exercises : If K is any subgroup of G containing H and such that H is normal in K, then KcN H • If K is a subgroup of N H ' then KH is a group and H is normal in KH. The normalizer of H is the largest subgroup of G in which H is normal.

By induction , we conclude that H m = {e }contains every 3-cycle, which is impossible, thus proving the theorem . Remark concerning the sign e(u). A priori, we defined the sign for a given n , so we should write cn(O") . However, suppose n < m . 3, so this restriction is equal to cn' Thus Am n S; = An. Next we prove some properties of the alternating group . (a) An is generated by the 3-cycles. Proof" Consider the product of two transpositions [ij][rs] . If they have an element in common, the product is either the identity or a 3-cycle.