Advanced Algebra Tutorial Exercise Set

Author: Wang Eofang
Publisher:
Publish Date: 2004-07-01
Features: Introduction This set of books, "Advanced Algebra Tutorial" (Volume 1 and Volume 2) and "Advanced Algebra Tutorial Exercise Book," is adapted from the highly acclaimed textbooks by Professor Wang Eofang of Peking University. It has been selected by the Beijing Higher Education Self-Study Examination Committee. The "Advanced Algebra Tutorial Exercise Book" summarizes the content of each chapter in "Advanced Algebra Tutorial" (Volume 1 and Volume 2) and provides solutions to the exercises. For calculation problems of the same type, one or two examples with detailed solutions are given, and proof problems are either fully proven or accompanied by hints. The supplementary problems at the end of the book offer more challenging exercises for readers, especially graduate students. Target Audience: Faculty and students of colleges and universities, self-study examinees in higher education, and radio and television university students.
Excerpt:
1.2n-Order Permutations
Content Summary
1. Some Definitions About n-Order Permutations
Definition 1 An ordered array composed of numbers 1, 2, ..., n is called an n-order permutation.
Definition 2 In a permutation, if a larger number appears before a smaller number, the two numbers are said to form an inversion. The total number of inversions in a permutation is called its inversion number, denoted by .
Definition 3 An n-order permutation with an even inversion number is called an even permutation, while one with an odd inversion number is called an odd permutation. The n-order permutation 12 is called the natural order permutation and is an even permutation.
2. Main Conclusions
Theorem 1 A transposition changes the parity of a permutation.
Corollary Any two n-order permutations can be transformed into each other through a series of transpositions. Moreover, if the two permutations have the same parity, the number of transpositions required is even; if they have opposite parities, the number of transpositions required is odd.