By Serge Lang

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 elemental innovations of algebra. For the revised 3rd version, the writer has further workouts and made quite a few corrections to the text.

Comments on Serge Lang's Algebra:

"Lang's Algebra replaced the best way graduate algebra is taught, conserving classical subject matters 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 used to be presented the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.

"The writer has a magnificent knack for offering the $64000 and fascinating rules of algebra in exactly 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**

To define j',, let xH be a coset of H . (x). This value is independent of the choice of coset representative x, and it is then trivially verified that j, is a homomorphism, is injective, and is the unique homomorphism satisfying our requirements. We shall say that j, is induced by [. can be factored into the following succession of homomorphisms : G s. 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.

Proof Let Fab(M) be the free abelian group generated by M. We denote the generator of Fab(M) corresponding to an element X E M by [x]. Let B be the subgroup generated by all elements of type [x + y] - [x] - [y] 40 I, §7 GROUPS where x, y EM. We let K(M) = Fab(M)/B , and let y : M --. K(M) be the map obtained by composing the injection of Minto Fab(M) given by x 1-+ [x], and the canonical map Fab(M) --. Fab(M)/B . It is then clear that y is a homomorphism, and satisfies the desired universal property.

The sum on the right is taken over the other orbits, and each index (G: Gx ) is then> 1, hence divisible by p. Since p divides the order of G, it follows that p divides the order of Z, hence in particular that G has a non-trivial center. Let a be an element of order p in Z, and let H be the cyclic group generated bya. Since H is contained in Z, it is normal. Letj" : G ~ G/H be the canonical map . Let pn be the highest power of p dividing (G: I). Then pn-l divides the order of G/H. Let K ' be a p-Sylow subgroup of G/H (by induction) and let K = f - l(K').