Understanding and Calculating the Inverse of a Permutation
Permutations are fundamental in mathematics, especially in combinatorics and group theory. One of the essential operations on permutations is finding their inverse. This article will guide you through the steps to find the inverse of a permutation, using a specific example to illustrate the process.
What is a Permutation?
A permutation is a one-to-one mapping or transformation from a finite set to itself. In simpler terms, it is a rearrangement of elements in a particular order. Let's denote a permutation as σ (sigma), where σ: n → n is a function from a finite set of n elements to itself.
Understanding the Permutation σ 1 2 5 7 3 4
The given permutation σ 1 2 5 7 3 4 can be represented in a two-row notation as follows:
(1 2 5 7 | 3 4)
This notation signifies that 1 maps to 2, 2 maps to 5, 5 maps to 7, 7 maps to 1, 3 maps to 4, and 4 maps to 3. These mappings can also be understood through disjoint cycles.
Step 1: Identifying Disjoint Cycles
The permutation can be broken down into disjoint cycles:
- The cycle (1 2 5 7) means 1 goes to 2, 2 goes to 5, 5 goes to 7, and 7 goes back to 1.
- The cycle (3 4) means 3 goes to 4, and 4 goes back to 3.
Step 2: Reversing the Mappings to Find the Inverse
To find the inverse σ-1, we reverse the direction of the mappings in each cycle:
For the cycle (1 2 5 7): - 2 goes to 1 - 5 goes to 2 - 7 goes to 5 - 1 goes to 7
This gives us the cycle (7 5 2 1).
For the cycle (3 4): - 4 goes to 3 - 3 goes to 4
This gives us the cycle (4 3).
Step 3: Writing the Inverse Permutation
Combining both cycles, we get the inverse permutation σ-1 as:
σ-1 (7 5 2 1 4 3)
Additional Insights
Recall that for permutations, we have the property σ ° δ-1 δ-1 ° σ-1. In the given example, σ 1 2 5 7 and δ 3 4. reversing the cycles gives us: - σ-1 7 5 2 1 - δ-1 4 3
Thus, the product σ ° δ-1 can be calculated as follows:
σ ° δ-1 4 3 7 5 2 1
If you write the permutation in full form and cycle notation, you can also observe the fixed points and their positions:
This shows that 6 and 7 are fixed points, and the cycles are as we identified earlier.
The inverse permutation σ-1 is thus 1 7 5 2 3 4.
In summary, to find the inverse of a permutation, you reverse the direction of the mappings in each cycle. This process can be illustrated through two-row notation and cycle notation, as demonstrated above.