Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
In Table 12.5.1 on p. 135 in [Biggs] (Page 11 in the Cryptography compendium), the different cycle types of S3 are explained and their numbers counted. Make a similar table for S6.

In Table 12.5.1 on p. 135 in [Biggs] (Page 11 in the Cryptography compendium), the different cycle types of S3 are explained and their numbers counted. Make a similar table for S6.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.1: The Geometry And Algebra Of Vectors
Problem 23EQ
See similar textbooks
icon
Related questions
Question
12.5 Classification of permutations 135 Table 12.5.1 Example Number Туре --[1°] [1°2] [1?3] [12*] (14] [23] [5] id (12)(3)(4)(5) (123)(4)(5) (12)(34)(5) (1234)(5) (123)(45) (12345) 10 20 15 30 20 24 120 The classification of permutations by type is doubly uscful, because there is an alternative description of the classes which has useful consequences in the alge- braic theory of permutations (Chapters 21 and 27). Let a and ß be permutations in Sn. If there is a permutation o in S, such that oao- = B, then we say that a and ß are conjugate. Theorem 12.5 Two permutations are conjugate if and only if they have the same type. Proof Suppose a and ß conjugate, so that oao= = B. If ¤(x¡) = x2, put y1 = o(x1), y2 = o(x2); if a(x2) = x3 put y3 = o(x3); and so on (Fig. 12.5). Then we have B(y1) = oa o-(o(x)) = oa(x1) = o(x2) = y2. X2 Fig. 12.5 Conjugate permutations. Similarly, B(y2) = y3, B(y3) = y4, and so on. Consequently, for each cycle (x,x2 ...x,) of a there is a corresponding cycle (yı2 ... Y;) of ß, and it follows that a and B have the same type. Conversely, suppose a and ß have the same type. Since they have the same number of cycles of each length, we can set up a bijective correspondence between their cycles, in which a typical cycle (x1x2...x,) of a will correspond to a cycle (212... z,) of ß. Let us define o by the rule o(x;) = z (1sisr), and use similar rules for the other cycles. Then oao- = B. since
Transcribed Image Text:12.5 Classification of permutations 135 Table 12.5.1 Example Number Туре --[1°] [1°2] [1?3] [12*] (14] [23] [5] id (12)(3)(4)(5) (123)(4)(5) (12)(34)(5) (1234)(5) (123)(45) (12345) 10 20 15 30 20 24 120 The classification of permutations by type is doubly uscful, because there is an alternative description of the classes which has useful consequences in the alge- braic theory of permutations (Chapters 21 and 27). Let a and ß be permutations in Sn. If there is a permutation o in S, such that oao- = B, then we say that a and ß are conjugate. Theorem 12.5 Two permutations are conjugate if and only if they have the same type. Proof Suppose a and ß conjugate, so that oao= = B. If ¤(x¡) = x2, put y1 = o(x1), y2 = o(x2); if a(x2) = x3 put y3 = o(x3); and so on (Fig. 12.5). Then we have B(y1) = oa o-(o(x)) = oa(x1) = o(x2) = y2. X2 Fig. 12.5 Conjugate permutations. Similarly, B(y2) = y3, B(y3) = y4, and so on. Consequently, for each cycle (x,x2 ...x,) of a there is a corresponding cycle (yı2 ... Y;) of ß, and it follows that a and B have the same type. Conversely, suppose a and ß have the same type. Since they have the same number of cycles of each length, we can set up a bijective correspondence between their cycles, in which a typical cycle (x1x2...x,) of a will correspond to a cycle (212... z,) of ß. Let us define o by the rule o(x;) = z (1sisr), and use similar rules for the other cycles. Then oao- = B. since
In Table 12.5.1 on p. 135 in [Biggs] (Page 11 in the Cryptography compendium), the different cycle types of S, are explained and their numbers counted. Make a similar table for S6.
Transcribed Image Text:In Table 12.5.1 on p. 135 in [Biggs] (Page 11 in the Cryptography compendium), the different cycle types of S, are explained and their numbers counted. Make a similar table for S6.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS