Difficult Question on Permutations

  • Context: Undergrad 
  • Thread starter Thread starter PhysicsHelp12
  • Start date Start date
  • Tags Tags
    Permutations
Click For Summary

Discussion Overview

The discussion revolves around determining the minimum positive integer k such that a given permutation, expressed in disjoint cycle notation, raised to the power of k equals the identity permutation. The scope includes theoretical reasoning about permutations and their properties.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asks for the minimum positive integer k such that t^k equals the identity, where t is a permutation in disjoint cycle notation.
  • Another participant prompts reasoning about the effects of multiplying the permutation τ by itself and suggests exploring τ^3 to understand the general behavior of repeated multiplication.
  • A third participant notes that disjoint cycles commute, stating that if τ is a product of disjoint cycles, then τ^n can be expressed as the product of each cycle raised to the power of n.
  • A later reply expresses a desire for a specific participant to arrive at the conclusion independently, indicating a preference for self-discovery in the reasoning process.

Areas of Agreement / Disagreement

Participants present various perspectives on the properties of permutations and their powers, but there is no consensus on the specific value of k or the implications of the multiplication of τ.

Contextual Notes

The discussion does not clarify certain assumptions about the structure of the permutation or the specific cycles involved, which may affect the determination of k.

Who May Find This Useful

Individuals interested in combinatorial mathematics, particularly those studying permutations and their properties in abstract algebra.

PhysicsHelp12
Messages
58
Reaction score
0
If you have a permutation writtein disjoint cycle notation ( I attatched it )

what's the minimum positive integer k such that t^k =Identity

t is tau
 

Attachments

  • Permutation.JPG
    Permutation.JPG
    5.2 KB · Views: 433
Physics news on Phys.org
Can you reason what will happen when \tau is multiplied by itself? And if you calculate \tau^3?
What will happen in general when we keep multiplying tau by itself?
 
Remember that disjoint cycles commute. Thus, if \tau = \tau_1 \tau_2 \dotsc \tau_m is a product of disjoint cycles, then \tau^n = \tau_1^n \tau_2^n \dotsc \tau_m^n.
 
Last edited:
Very well adrian, although I was sort of hoping PhysicsHelp12 would figure that out by himself.
 

Similar threads

Undergrad About permutation acting on the Identity matrix
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Cycles from patterns in a permutation matrix
  • · Replies 3 ·
Replies
3
Views
2K
High School Determinant of a transposed matrix
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Subgroup axioms for a symmetric group
  • · Replies 1 ·
Replies
1
Views
1K
Graduate What Is the Role of 'n' and the Virgule in the Wedge Product Formalism?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
2K
High School Is it invalid to redefine the sgn function in this way in a proof?
  • · Replies 15 ·
Replies
15
Views
5K
Undergrad Prove Even # Transpositions in Identity Permutation w/ Induction & Contradiction
  • · Replies 8 ·
Replies
8
Views
5K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Can't understand a step in an LU decomposition proof
  • · Replies 1 ·
Replies
1
Views
3K