By Serge Lang

ISBN-10: 146130041X

ISBN-13: 9781461300410

This e-book 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 execs who use higher-level algebra. It effectively addresses the fundamental suggestions of algebra. For the revised 3rd version, the writer has extra workouts and made quite a few corrections to the text.

Comments on Serge Lang's Algebra:

"Lang's Algebra replaced the way in which graduate algebra is taught, protecting classical themes yet introducing language and methods of considering from class thought and homological algebra. It has affected all next graduate-level algebra books."

-April 1999 Notices of the AMS, asserting that the writer was once provided the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.

"The writer has a powerful knack for offering the real and engaging 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 components of algebra."

-MathSciNet's assessment of the 1st version

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**Extra resources for Algebra**

**Example text**

A cyclic group) is abelian (resp. cyclic). A refinement of a tower G = Go => GI => • . => Gm is a tower which can be obtained by inserting a finite number of subgroups in the given tower. e. Gm = {e} in the above notation). 1. Let G be a finite group. An abelian tower of G admits a cyclic refinement. Let G be a finite solvable group. Then G admits a cyclic tower whose last element is {e}. Proof The second assertion is an immediate consequence of the first, and it clearly suffices to prove that if G is finite, abelian, then G admits a cyclic tower ending with {e}.

If n ~ 5 then the alternating group An is simple. Proof. Let N be a non-trivial normal subgroup of An' We prove that N contains some 3-cycle , whence the theorem follows by (b). Let 0" EN , 0" id, be an element which has the maximal number of fixed points; that is , integers i such that O"(i) = i. It will suffice to prove that 0" is a 3-cycle or the identity. Then some orbits have more than one element. Suppose all orbits have 2 elements (except for the fixed points) . Since 0" is even, there are at least two such orbits.

7 . In general, an operation of G on S is said to be transitive if there is only one orbit. Examples. The symmetric group Sn operates transitively on {I, 2, . . , n}. (See p. 1 of Chapter VII, we shall see a non-trivial example of transitive action of a Galois group operating on the primes lying above a given prime in the ground ring. In topology, suppose we have a universal covering space p : X' -. X , where X is connected. Given x E X , the fundamental group 7l'1 (X) operates transitively on the inverse image p -l (x).

### Algebra by Serge Lang

by Ronald

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