Even Permutations Calculator
Determine if a permutation is even or odd by decomposing it into transpositions
Tip: Enter your permutation in either one-line or cycle notation (e.g., "2 3 1 5 4" or "(1 2 3)(4 5)")
Permutation Analysis Result
Permutation:
Cycle Notation:
Number of Inversions:
Parity:
Transposition Decomposition
An even permutation can be expressed as an even number of transpositions (swaps of two elements)
Inversion Pairs
An inversion is a pair (i,j) where i < j but σ(i) > σ(j):
Understanding Permutation Parity
In group theory, the parity of a permutation refers to whether it can be expressed as an even or odd number of transpositions (swaps of two elements).
Key Concepts
Even Permutation
Can be expressed as an even number of transpositions. Forms the alternating group Aₙ, a subgroup of the symmetric group Sₙ.
Example: (1 2 3) = (1 2)(1 3)
Odd Permutation
Requires an odd number of transpositions. The product of two odd permutations is even.
Example: (1 2) [single transposition]
Determining Parity
There are several equivalent ways to determine if a permutation is even or odd:
- Transposition Count: Decompose into transpositions and count them
- Inversion Count: Count pairs (i,j) where i < j but σ(i) > σ(j)
- Cycle Structure: For a cycle of length k, it can be written as k-1 transpositions
Theorem: Every permutation is either even or odd, but never both. The identity permutation is even (can be considered as 0 transpositions).
Applications
- Determining solvability of the 15 puzzle
- Analyzing Rubik's cube configurations
- Calculating determinants in linear algebra
- Group theory and abstract algebra
Frequently Asked Questions
What's the difference between even and odd permutations?
An even permutation can be expressed as an even number of transpositions (element swaps), while an odd permutation requires an odd number. The identity permutation is even (0 transpositions). The product of two even permutations is even, while the product of an even and odd permutation is odd.
How can I tell if a permutation is even without decomposing it?
You can count inversions (pairs where a larger number precedes a smaller one) - if the count is even, the permutation is even. Alternatively, for cycle notation: a permutation is even if it has an even number of even-length cycles (when considering the total number of transpositions in its decomposition).
Why is permutation parity important?
Parity is fundamental in group theory - the even permutations form the alternating group, a subgroup of the symmetric group. It determines solvability of puzzles like the 15-puzzle, affects determinant calculations, and appears in physics (fermions vs bosons). Many algorithms check parity to validate configurations.
Can a permutation be both even and odd?
No. A fundamental theorem states that while a permutation can be decomposed into transpositions in multiple ways, the parity (evenness or oddness) of the number of transpositions will always be the same for a given permutation.
How does cycle length affect parity?
A cycle of length k can be written as k-1 transpositions. Thus odd-length cycles are even permutations (since k-1 is even when k is odd) and even-length cycles are odd permutations. The parity of the whole permutation is the sum of these contributions modulo 2.
Practice Problems
Basic Problems
- (1 2 3 4)
- (1 2)(3 4 5)
- (1 3 5)(2 4 6)
Advanced Problems
- (1 2 3)(4 5 6)(7 8 9)
- (1 2 3 4 5 6)
- (1 2)(3 4)(5 6)(7 8)
Challenge: Before checking, try to determine the parity by: (1) counting transpositions in your head, (2) counting inversions, and (3) examining cycle structure. Verify all methods give the same result.