Product of Permutations, Permutation Cycles and Transpositions
Applying one Transposition / Permutation Cycle / Permutation after another is equated as calculating Product of Transpositions / Permutation Cycles / Permutation. Product of Transpositions, Permutation Cycles, or in general Permutations give rise to another Permutation.
This document details all the ways in which such mutiplications are carried out and the rules used to perform such multiplications.
The Transpositions / Permutation Cycles / Permutations always get applied from Left to the Right (i.e. the Left Most always gets applied first follwed by the One to the Right of it and so on).
If \(P\), \(Q\) and \(R\) are any 3 Transpositions / Permutation Cycles / Permutations then
\(P\cdot Q=\)First Applying the Transposition / Permutation Cycle / Permutation \(P\) and then applying Transposition / Permutation Cycle / Permutation \(Q\)
\(Q\cdot P=\)First Applying the Transposition / Permutation Cycle / Permutation \(Q\) and then applying Transposition / Permutation Cycle / Permutation \(P\)
\(P\cdot Q \cdot R=\)First Applying the Transposition / Permutation Cycle / Permutation \(P\), then applying Transposition / Permutation Cycle / Permutation \(Q\) and then applying Transposition / Permutation Cycle / Permutation \(R\)
Multiplication of Transpositions / Permutation Cycles / Permutations is Commutative only when they are Disjoint i.e they have No Elements in Common. If \(P\) and \(Q\) are any 2 Disjoint Transpositions / Permutation Cycles / Permutations then
\(P\cdot Q=Q\cdot P\)
Multiplication of Transpositions / Permutation Cycles / Permutations is Associative. If \(P\), \(Q\) and \(R\) are any 3 Transpositions / Permutation Cycles / Permutations then
\(P \cdot (Q\cdot R)=(P\cdot Q)\cdot R\)
Product of 2 Transpositions: The following 3 cases arise when Multiplying 2 Transpositions
Both Transpositions Have No Common Element. In such cases, the Order in which the two Transpositions get multiplied Do Not Matter.
Multiplying such two Transpositions result in Permutations having 2 Disjoint Permutation Cycles, each of Cycle Length 2. The following gives an example
Both Transpositions Have One Common Element. In such cases, the Order in which the two Transpositions get multiplied Do Matter.
Multiplying such two Transpositions result in Permutations having a Permutation Cycle of Cycle Length 3. The \(1^{st}\) Element of this Resultant Permutation Cycle is the Common Element.
The \(2^{nd}\) Element is the Uncommon Element of the \(1^{st}\) Transposition. The \(3^{rd}\) Element is the Uncommon Element of the \(2^{nd}\) Transposition. The following gives an example
Both Transpositions Have Same Elements. Multiplying such two Transpositions result in Identity Permutations having Permutation Cycles of Cycle Length 0.
The following gives an example
\((2\hspace{.1cm}5)(5\hspace{.1cm}2)=()\) ...(3)
Please Note that the Permutation Cycles of Cycle Length 0 are Never Explicitly Written.
Product of a Transposition and a Permutation Cycle Having Cycle Length \(> 2\): The following 4 cases arise when Multiplying a Transposition and a Permutation Cycle Having Cycle Length \(> 2\)
Transposition and the Permutation Cycle Have No Common Element. In such cases, the Order in which the Transposition and the Permutation Cycle get multiplied Do Not Matter.
Such Multiplications result in Permutations having 2 Disjoint Permutation Cycles, One of the Transposition and Other of the Permutation Cycle. The following gives an example
Transposition and the Permutation Cycle Have One Common Element. In such cases, the Order in which the Transposition and the Permutation Cycle get multiplied Do Matter.
Such Multiplications result in Permutations with a Permutation Cycle whose Cycle Length is 1 More than that of existing Permutation Cycle.
If the Transposition is Applied Before the Permutation Cycle, the Uncommon Element gets Added After the Common Element in the Permutation Cycle. The following gives an example
In the equation (5) above, since the Transposition \((4\hspace{.1cm}8)\) gets applied before the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\),
the Uncommon Element \(4\) gets placed after \(8\) in the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\) to give the Resultant
Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}4\hspace{.1cm}3)\).
If the Transposition is Applied After the Permutation Cycle, the Uncommon Element gets Added Before the Common Element in the Permutation Cycle. The following gives an example
In the equation (6) above, since the Transposition \((4\hspace{.1cm}8)\) gets applied after the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\),
the Uncommon Element \(4\) gets placed before \(8\) in the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\) to give the Resultant
Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}4\hspace{.1cm}8\hspace{.1cm}3)\).
Both Elements of Transposition are Adjacent in the Permutation Cycle. In such cases, the Order in which the Transposition and the Permutation Cycle get multiplied Do Matter.
Such Multiplications result in Permutations with a Permutation Cycle whose Cycle Length is 1 Less than that of existing Permutation Cycle.
If the Transposition is Applied Before the Permutation Cycle, the Common Element on the Right in the Permutation Cycle gets removed from the Permutation Cycle. The following gives an example
In the equation (7) above, since the Transposition \((3\hspace{.1cm}8)\) gets applied before the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\),
the Common Element \(3\) which is present on the Right of Common Element \(8\) in the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\) gets removed to give the Resultant
Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8)\).
If the Transposition is Applied After the Permutation Cycle, the Common Element on the Left in the Permutation Cycle gets removed from the Permutation Cycle. The following gives an example
In the equation (8) above, since the Transposition \((3\hspace{.1cm}8)\) gets applied after the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\),
the Common Element \(8\) which is present on the Left of Common Element \(3\) in the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}3)\) gets removed to give the Resultant
Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}3)\).
Both Elements of Transposition are Not-Adjacent in the Permutation Cycle. In such cases, the Order in which the Transposition and the Permutation Cycle get multiplied Do Matter.
Such Multiplications cause Splitting of the Permutation Cycle resulting in Permutations with 2 Disjoint Permutation Cycles..
If the Transposition is Applied Before the Permutation Cycle, the Split Occurs in the Permutation Cycle After the Elements that it has Common with the Transposition. The following gives an example
In the equation (9) above, since the Transposition \((3\hspace{.1cm}8)\) gets applied before the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}7\hspace{.1cm}5\hspace{.1cm}3\hspace{.1cm}6)\),
the Split Occurs in the Permutation Cycle After Common Elements \(8\) and \(3\) to give the Resultant Disjoint Permutation Cycles \((6\hspace{.1cm}2\hspace{.1cm}9\hspace{.1cm}8)(7\hspace{.1cm}5\hspace{.1cm}3)\).
If the Transposition is Applied After the Permutation Cycle, the Split Occurs in the Permutation Cycle Before the Elements that it has Common with the Transposition. The following gives an example
In the equation (10) above, since the Transposition \((3\hspace{.1cm}8)\) gets applied after the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}7\hspace{.1cm}5\hspace{.1cm}3\hspace{.1cm}6)\),
the Split Occurs in the Permutation Cycle Before Common Elements \(8\) and \(3\) to give the Resultant Disjoint Permutation Cycles \((3\hspace{.1cm}6\hspace{.1cm}2\hspace{.1cm}9)(8\hspace{.1cm}7\hspace{.1cm}5)\).
Product of 2 Permutations Cycles Each Having Cycle Length \(> 2\): The following 2 cases arise when Multiplying 2 Permutations Cycles Each Having Cycle Length \(> 2\)
Both Permutation Cycles Have No Common Element. In such cases, the Order in which the two Permutation Cyles get multiplied Do Not Matter.
Multiplying such two Permutation Cycles result in Permutations having 2 Disjoint Permutation Cycles. The following gives an example
Both Permutation Cycles Have One or More Elements in Common. In such cases, the Order in which the two Permutation Cyles get multiplied Do Matter.
Such Multiplications are carried out by Decomposing One of the Permutation Cycles into Transpositions and then Progressively Mutiplying the Other Permutation Cycle with the Transpositions. The following gives an example
The following Calculates the Product of Permutation Cycles given in expression (12) above by Decomposing the Permutation Cycle \((6\hspace{.1cm}5\hspace{.1cm}7\hspace{.1cm}9\hspace{.1cm}1)\) into Transpositions
The following Calculates the Product of Permutation Cycles given in expression (12) above by Decomposing the Permutation Cycle \((2\hspace{.1cm}9\hspace{.1cm}8\hspace{.1cm}6)\) into Transpositions
Product of 2 Permutations: Product of 2 Permutations can be carried out in the following 2 ways
Representing the Permutations in Form of Permutation Cycles and Mutiplying Them using all the Rules of Mutiplications as given above.
Representing the Permutations in Form of Permutation Tables and Mutiplying Them. The following example demonstrates
Multiplication of 2 Permutation Tables representing Permutations \(P\) and \(Q\) as given below
In the Permutation obtained through Product \(P\cdot Q\), the Permutation \(P\) gets applied before Permutation \(Q\). Following gives the Steps for this Multiplication
In Permutation \(P\) the Item at \(1^{st}\) Location gets mapped to \(4^{th}\) Location. But through Permutation \(Q\) (which is applied after Permutation \(P\)) the Item at \(4^{th}\) Location is then mapped to \(2^{nd}\) Location.
As a result, through Permutation \(P \cdot Q\) the Item at \(1^{st}\) Location gets mapped to \(2^{nd}\) Location.
In Permutation \(P\) the Item at \(2^{nd}\) Location stays at \(2^{nd}\) Location. But through Permutation \(Q\) (which is applied after Permutation \(P\)) the Item at \(2^{nd}\) Location is then mapped to \(1^{st}\) Location.
As a result, through Permutation \(P \cdot Q\) the Item at \(2^{nd}\) Location gets mapped to \(1^{st}\) Location.
In Permutation \(P\) the Item at \(3^{rd}\) Location gets mapped to \(1^{st}\) Location. But through Permutation \(Q\) (which is applied after Permutation \(P\)) the Item at \(1^{st}\) Location is then mapped to \(5^{th}\) Location.
As a result, through Permutation \(P \cdot Q\) the Item at \(3^{rd}\) Location gets mapped to \(5^{th}\) Location.
In Permutation \(P\) the Item at \(4^{th}\) Location gets mapped to \(6^{th}\) Location. But through Permutation \(Q\) (which is applied after Permutation \(P\)) the Item at \(6^{th}\) Location is then mapped to \(4^{th}\) Location.
As a result, through Permutation \(P \cdot Q\) the Item at \(4^{th}\) Location remains at \(4^{th}\) Location.
In Permutation \(P\) the Item at \(5^{th}\) Location stays at \(5^{th}\) Location. But through Permutation \(Q\) (which is applied after Permutation \(P\)) the Item at \(5^{th}\) Location is then mapped to \(3^{rd}\) Location.
As a result, through Permutation \(P \cdot Q\) the Item at \(5^{th}\) Location gets mapped to \(3^{rd}\) Location.
In Permutation \(P\) the Item at \(6^{th}\) Location gets mapped to \(3^{rd}\) Location. But through Permutation \(Q\) (which is applied after Permutation \(P\)) the Item at \(3^{rd}\) Location is then mapped to \(6^{th}\) Location.
As a result, through Permutation \(P \cdot Q\) the Item at \(6^{th}\) Location remains at \(6^{th}\) Location.
In the Permutation obtained through Product \(Q\cdot P\), the Permutation \(Q\) gets applied before Permutation \(P\). Following gives the Steps for this Multiplication
In Permutation \(Q\) the Item at \(1^{st}\) Location gets mapped to \(5^{th}\) Location. But through Permutation \(P\) (which is applied after Permutation \(Q\)) the Item at \(5^{th}\) Location stays at \(5^{th}\) Location.
As a result, through Permutation \(Q \cdot P\) the Item at \(1^{st}\) Location gets mapped to \(5^{th}\) Location.
In Permutation \(Q\) the Item at \(2^{nd}\) Location gets mapped to \(1^{st}\) Location. But through Permutation \(P\) (which is applied after Permutation \(Q\)) the Item at \(1^{st}\) Location is then mapped to \(4^{th}\) Location.
As a result, through Permutation \(Q \cdot P\) the Item at \(2^{nd}\) Location gets mapped to \(4^{th}\) Location.
In Permutation \(Q\) the Item at \(3^{rd}\) Location gets mapped to \(6^{th}\) Location. But through Permutation \(P\) (which is applied after Permutation \(Q\)) the Item at \(6^{th}\) Location is then mapped to \(3^{rd}\) Location.
As a result, through Permutation \(Q \cdot P\) the Item at \(3^{rd}\) Location stays at \(3^{rd}\) Location.
In Permutation \(Q\) the Item at \(4^{th}\) Location gets mapped to \(2^{nd}\) Location. But through Permutation \(P\) (which is applied after Permutation \(Q\)) the Item at \(2^{nd}\) Location stays at \(2^{nd}\) Location.
As a result, through Permutation \(Q \cdot P\) the Item at \(4^{th}\) Location remains at \(2^{nd}\) Location.
In Permutation \(Q\) the Item at \(5^{th}\) Location gets mapped to \(3^{rd}\) Location. But through Permutation \(P\) (which is applied after Permutation \(Q\)) the Item at \(3^{rd}\) Location is then mapped to \(1^{st}\) Location.
As a result, through Permutation \(Q \cdot P\) the Item at \(5^{th}\) Location gets mapped to \(1^{st}\) Location.
In Permutation \(Q\) the Item at \(6^{th}\) Location gets mapped to \(4^{th}\) Location. But through Permutation \(P\) (which is applied after Permutation \(Q\)) the Item at \(4^{th}\) Location is then mapped to \(6^{th}\) Location.
As a result, through Permutation \(Q \cdot P\) the Item at \(6^{th}\) Location remains at \(6^{th}\) Location.