Finite Groups: An Introduction

Description

Finite group theory is a topic remarkable for the simplicity of its statements and the difficulty of their proofs. It is used in an essential way in several branches of mathematics—for instance, in number theory.

This book is a short introduction to the subject, written both for beginners and for mathematicians at large. There are ten chapters: Preliminaries, Sylow theory, Solvable groups and nilpotent groups, Group extensions, Hall subgroups, Frobenius groups, Transfer, Characters, Finite subgroups of GL_n, and Small groups.

Each chapter is followed by a series of exercises.

About the Author

Jean-Pierre Serre attended the École Normale Supérieure (1945–48) and the Sorbonne (Ph.D., 1951), both now part of the Universities of Paris. Between 1948 and 1954 he was at the National Center for Scientific Research in Paris, and after two years at the University of Nancy he returned to Paris for a professorship at the Collège de France. He retired in 1994. Between 1983 and 1986 he served as vice president of the International Mathematical Union. Serre’s mathematical contributions, leading to a Fields Medal in 1954, were largely in the field of algebraic topology, but his later work ranged widely—within algebraic geometry, group theory, and especially number theory. By seeing unifying ideas, he helped to unite disparate branches of mathematics. One of the more recent phenomena to which he was a principal contributor was the application of algebraic geometry to number theory—applications now falling into a separate subclass called arithmetic geometry. Serre has published many books: Groupes algébriques et corps de classes (1959); Corps Locaux (1962); Cohomologie Galoisienne (1964); Lie Algebras and Lie Groups (1965); Algèbre locale, multiplicités (1965); Algèbres de Lie semi-simples complexes (1966); Représentations linéaires des groupes finis (1967); Abelian l-adic Representations and Elliptic Curves (1968); Cours d’arithmétique (1970); Arbres, amalgames, SL₂ (1977); Lectures on the Mordell-Weil Theorem (1989); Topics in Galois Theory (1992); Lectures on N_x(p) (2012). His Collected Works (1949–1984) were published in 1986, followed in 2000 by Collected Works (1985–1998). In 1995 Serre received a Leroy P. Steele Prize for A Course in Arithmetic, and in 2003 he was awarded the first Abel Prize by the Norwegian Academy of Science and Letters.

This volume is part of the Surveys of Modern Mathematics book series.

Publications