By Serge Lang

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 variation, the writer has further workouts 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, preserving classical subject matters yet introducing language and methods of pondering from type conception and homological algebra. It has affected all next graduate-level algebra books."

-April 1999 Notices of the AMS, saying that the writer was once provided 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 principles of algebra in precisely the "right" approach, and he by no means will get slowed down within the dry formalism which pervades a few elements of algebra."

-MathSciNet's overview of the 1st version

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**Example text**

If a has finite period, and A is simple, then A = (a) . Let n be the period and suppose n not prime. Write n = rs with r, s > 1. Then a' =1= e and a' generates a proper subgroup, contradicting the simplicity of A, so a has prime period and A is cyclic of order p. * Examples. 2 of Chapter XIII (PSLn(F) , a group of matrices to be defined in that chapter) . Since a non-cyclic simple group is not solvable, we get thereby examples of non-solvable groups. A major program of finite group theory is the classification of all finite simple groups.

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). Then x t-+ nx is a homomorphism of G into Perm(S) . So an operation of G on S could also be defined as a mapping G x S --+ S satisfying the above two properties.

Our next task is to desc ribe the structure of finite abelian p-groups. Let ~ I. A finite p-group A is said to be of type (pr" ,pro) if A is isomorphic to the product of cyclic groups of orders p" (i = 1, , s). We shall need the following remark . r 1 , ••• , rs be integers Remark. Let A be a finite abelian p-group . Let b be an element of A, b 0 . Let k be an integer ~ such that pkb 0, and let pm be the period of p'b , Then b has period pk+m . 2 . Every finite abelian p-group is isomorphic to a product of cyclic p-groups.