FlickS said:Homework Statement
Let α = (α1α2...αs) be a cycle, for positive integers α1α2...αs. Let π be any permutation that παπ-1 is the cycle (π(α1)πα2...π(αs)).
Homework Equations
The Attempt at a Solution
I started by choosing a specific α and π, and tried finding παπ-1 to give myself some idea of what to do but have had no luck. Suggestions would be welcomed.
FlickS said:I would get παπ−1(π(α1)) = πα(α1)) = πα2?
It gives me the next element in the cycle. So παπ−1 would be that cycle.
I'm still relatively confused.
Okay, that definitely makes its more clear. Thanks so much!Dick said:Let's set ##\sigma=\pi \alpha \pi^{-1}## for short. You've shown ##\sigma(\pi(\alpha_1))=\pi(\alpha_2)##. Generalizing that I'd say the cycle structure of ##\sigma## is ##(\pi(\alpha_1)\pi(\alpha_2)...)##. Still confused?
Cycle decomposition of permutations is a method of breaking down a permutation into its constituent cycles. A permutation is a way of rearranging a set of objects, and a cycle is a specific type of rearrangement where each object is moved to the position of the next object in the cycle.
Cycle decomposition is important in mathematics because it provides a way to analyze and understand permutations. By breaking down a permutation into cycles, we can better understand its structure and properties, and use it to solve problems in various fields of mathematics, such as group theory and combinatorics.
Cycle decomposition is typically represented using a visual diagram called a cycle notation. In this notation, each cycle is written within parentheses, with the elements in the cycle listed in the order in which they are moved. For example, the permutation (1 2 3)(4 5) would be represented as (1 2 3)(4 5) in cycle notation.
There are several properties of cycle decomposition, including the fact that every permutation can be decomposed into cycles, and that the order of the cycles does not affect the overall permutation. Additionally, the length of a cycle (i.e. the number of elements in the cycle) is equal to the number of times that cycle is repeated in the permutation.
Cycle decomposition can be used to solve problems involving permutations, such as finding the order of a permutation or determining whether two permutations are equivalent. It can also be used to find the inverse of a permutation, and to study the structure of groups of permutations.