Difficult Question on Permutations

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In summary, the conversation is about finding the minimum positive integer k for which t^k = Identity, where t is represented in disjoint cycle notation. It is mentioned that when t is multiplied by itself, it results in \tau^3 and in general, when t is multiplied by itself multiple times, it follows the rule that \tau^n = \tau_1^n \tau_2^n \dotsc \tau_m^n, as disjoint cycles commute. The conversation ends with the expectation that PhysicsHelp12 would have figured this out on his own.
  • #1
PhysicsHelp12
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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
 

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  • #2
Can you reason what will happen when [itex]\tau[/itex] is multiplied by itself? And if you calculate [itex]\tau^3[/itex]?
What will happen in general when we keep multiplying tau by itself?
 
  • #3
Remember that disjoint cycles commute. Thus, if [itex]\tau = \tau_1 \tau_2 \dotsc \tau_m[/itex] is a product of disjoint cycles, then [itex]\tau^n = \tau_1^n \tau_2^n \dotsc \tau_m^n[/itex].
 
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  • #4
Very well adrian, although I was sort of hoping PhysicsHelp12 would figure that out by himself.
 

Related to Difficult Question on Permutations

What is a permutation?

A permutation is a way of arranging a set of objects in a particular order. It is essentially a different ordering of the same set of objects.

What is the difference between a permutation and a combination?

A permutation is an ordered arrangement of objects, while a combination is an unordered selection of objects. Permutations take into account the order in which the objects are arranged, while combinations do not.

How do I calculate the number of permutations?

The number of permutations can be calculated using the formula n!/(n-r)!, where n is the total number of objects and r is the number of objects being arranged.

What is the significance of permutations in mathematics?

Permutations are used in various fields of mathematics, including probability, statistics, and combinatorics. They are used to solve problems involving arrangements, such as counting the number of possible outcomes in a given scenario.

What are the practical applications of permutations?

Permutations have many practical applications, such as in cryptography, where they are used to create secure codes. They are also used in computer science for data encryption and compression. In addition, permutations are used in music theory to create different musical arrangements.

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