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Dr. Thomas Madsen
Abstract
Within Group Theory, there is often a lack of rigor when defining the fundamental element of the permutation group. Definitions by example and explanation are prevalent yet lacking. This project seeks to define the cycle with the same level of mathematical rigor as other constructs. In exploring possible definitions, two equivalent statements are discovered and proved equivalent. Having these two definitions allow certain theorems to be proved more easily as it's been shown that:
σ ∈ Sn is an m-cycle ⇔ Xσ = {σ(a)^i | 0 ≤ i < m} ∀a ∈ Xσ
Some Classical Definitions
In Abstract Algebra and Group Theory there are objects called groups. These objects are useful to many researchers. In my project Defining the Cycle, I use the following definitions.
A Group is a set of elements with a binary operation that follows these 4 group axioms:
(Closure) For any combination of two elements from the initial set, the operation maps those two elements to another element in the set.
(Associativity) For any combination of three elements from the initial set, any two applications of the operation yield the same element in return.
(Identity) There exists an element in the set for which applying the operation to that element and any other element in the set, the result is the other element.
(Inverses) For every element in the set, there exists another element in the set for which the operation returns an element described in 3.
The order of an element in a group is the smallest number of iterations applying the operation to the given element and itself to yield an element as described in 3.
The Permutation Group (Sn) is the set of bijective functions (permutations) from a given set to itself and the operation of function composition. For the purposes of this project, the underlying set for which the functions act on is taken to be the set of natural numbers up to a given number (ℕn).
Original Work
Prelimenary Definitions
We define a set of fixed points under a permutation φ in Sn to be Yφ. Then Yφ has the form {y∈ℕn|φ(y)=y}. In a similar manner, we define a set of non-fixed points under φ to be Xφ. Xφ has the form {x∈ℕn|φ(x)=x}.
Cycle Definition
If the order of an element is equal to the order of the set of non-fixed points (say m), then we say the element is an m-cycle.
A 2-cycle is a transposition.
Two cycles are disjoint if their respective sets of non-fixed points are disjoint.
Theorems
Cycle and Cycle Decomposition
For two disjoint cycles, the resulting combination of the two cycles does not depend on which cycle is applied first. This is to say that disjoint cycles permute.
The order of the combination of two disjoint cycles is equal to the least common multiple of the order of the cycles themselves.
A cycle may not be written as the product of two or more disjoint cycles. This is to say that a cycle has some fundamental indecomposable nature to it.
Xφ can alternatively be defined by collecting the resulting elements from applying φ to a given element a successively a number of times equal to the order of φ. This is to say Xφ={φ^i(a)|0<i≤m} where |φ|=m.
Any permutation with two non-fixed points is a transposition.
Any non-identity permutation may be written as the composition of disjoint cycles.
Any permutation can be decomposed into a unique set of disjoint cycles.
Transpositions
As a note, these theorems follow the classical literature though the proofs of which use the original definitions discussed above.
Any permutation may be written as the composition of transpositions.
A transposition is a self-inverse
Any transposition cannot be written as the composition of two transpositions.
A permutation that can be written as the composition of an even number of transpositions (an even permutation) cannot be written as the composition of an odd number of transpositions. In a similar vein, a permutation that can be written as the composition of an odd number of transpositions (an odd permutation) cannot be written as the composition of an even number of transpositions.
The number of even permutations is equal to half the total number of permutations.
The set of even permutations is a group under function composition.
MathFest 2021
I presented the findings in this project at the Pi Mu Epsilon session at MathFest 2021. Due to time constraints, I presented only on the two equivalent, rigorous criterion for a cycle. The focus in this presentation is showing the equivalency and showing that they both fit in with the classical applications of cycles.
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