Finite Group Theory
نویسنده
چکیده
In the preface of Finite Group Theory the author, I. Martin Isaacs, states that his principal reason for writing the book was to expose students to the beauty of the subject. Finite group theory is indeed a subject which has both beautiful theory and beautiful examples. The simplicity and elegance of the group axioms have made group theory an almost universal choice as a starting point in the teaching of abstract mathematics. But Isaacs had more than this in mind. Throughout the history of the subject there have been many examples of theorems with proofs which are ingenious, highly original, or which establish an important new general principle, proofs of great aesthetic value. Everyone who has taken a graduate algebra course is aware of Sylow’s theorems, and of the noticable increase in depth of the discussion which follows them. A slightly less well known example is Frattini’s proof that the intersection of all maximal subgroups (now called the Fratttini subgroup in his honor) must be nilpotent. This short proof introduced two basic mathematical principles. One is the key algebraic concept of a radical, which also underlies many fundamental results such as Nakayama’s Lemma on commutative rings. The other is the importance of transitive group actions and, specifically, the method of applying Sylow’s theorems which has become known as the Frattini Argument. Another classical theorem whose proof has an almost magical quality is Burnside’s pq theorem, which states that a group whose order is divisible by at most two primes must be solvable. Its wonderful proof was one of the earliest major applications of character theory. We will return to it later. The first half of the twentieth century witnessed the definitive work of P. Hall, whose generalizations of Sylow’s theorems illuminated the structure of solvable finite groups. The modern era of finite group theory began around 1959, when a number of startlingly original and powerful ideas were introduced by Thompson, first to prove a conjecture of Frobenius, then the Feit-Thompson Odd Order Theorem and the classification of the N-groups, which include all simple groups with the property that every proper subgroup is solvable. These results caused an explosion of research leading eventually to the classification
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