Even Permutations Calculator
Result:
Calculate the number of even permutations in a set. Essential for understanding group theory, alternating groups, and permutation parity in mathematical studies.
What are Even Permutations?
Even permutations are permutations that can be expressed as an even number of transpositions (swaps). For any set of n elements (n ≥ 2), exactly half of all permutations are even.
Even Permutation Formula
Even permutations = n!/2 (for n ≥ 2)Odd permutations = n!/2 (for n ≥ 2)Special case: n=1 has 1 even permutation
Understanding Permutation Parity
⚖️ Parity Concept
Every permutation can be classified as even or odd:
Even Permutations:
- Can be written as an even number of swaps
- Have permutation sign +1
- Form the alternating group A_n
- Preserve orientation in geometry
- Count: n!/2 for n ≥ 2
Odd Permutations:
- Require an odd number of swaps
- Have permutation sign -1
- Don't form a group (no identity)
- Reverse orientation in geometry
- Count: n!/2 for n ≥ 2
Examples by Set Size
All 3! = 6 permutations classified by parity:
Even Permutations (3):
- (1)(2)(3) - identity (0 swaps)
- (1 2 3) - 3-cycle (2 swaps)
- (1 3 2) - 3-cycle (2 swaps)
Odd Permutations (3):
- (1 2) - transposition (1 swap)
- (1 3) - transposition (1 swap)
- (2 3) - transposition (1 swap)
Result: 3 even, 3 odd (equal split)
For n = 4, we have 4! = 24 total permutations:
Even permutations = 24/2 = 12
Odd permutations = 24/2 = 12
Examples of even permutations:
- Identity: (1)(2)(3)(4) - 0 transpositions
- Double transposition: (1 2)(3 4) - 2 transpositions
- 4-cycle: (1 2 3 4) - 3 transpositions
Examples of odd permutations:
- Single transposition: (1 2) - 1 transposition
- 3-cycle: (1 2 3) - 2 transpositions
Applications in Mathematics
Group Theory
- Alternating groups A_n
- Simple group classification
- Normal subgroups
- Group homomorphisms
- Galois theory applications
Geometry
- Orientation preservation
- Determinant calculations
- Coordinate transformations
- Symmetry operations
- Crystallography
Linear Algebra
- Matrix determinants
- Eigenvalue calculations
- Characteristic polynomials
- Volume computations
- Change of basis
Common Values
| n | Total (n!) | Even | Odd | Alternating Group |
|---|---|---|---|---|
| 1 | 1 | 1 | 0 | A₁ = {e} |
| 2 | 2 | 1 | 1 | A₂ = {e} |
| 3 | 6 | 3 | 3 | A₃ = C₃ |
| 4 | 24 | 12 | 12 | A₄ |
| 5 | 120 | 60 | 60 | A₅ (simple) |
| 6 | 720 | 360 | 360 | A₆ |
| 7 | 5040 | 2520 | 2520 | A₇ (simple) |
| 8 | 40320 | 20160 | 20160 | A₈ |
Determining Permutation Parity
🔍 Methods to Check Parity
Transposition Count Method:
- Express permutation as product of transpositions
- Count the number of transpositions
- Even count → even permutation
- Odd count → odd permutation
Cycle Structure Method:
- Write permutation in cycle notation
- k-cycle = k-1 transpositions
- Sum all (cycle length - 1)
- Even sum → even permutation
Frequently Asked Questions (FAQ)
This follows from the properties of the alternating group:
- Symmetry: There's a bijection between even and odd permutations
- Mapping: Composing any permutation with a transposition changes its parity
- Example: If σ is even, then σ∘(1 2) is odd
- Result: This creates a one-to-one correspondence
- Exception: n=1 has only the identity (even), so 1 even, 0 odd
The alternating group A_n consists of all even permutations:
- Definition: A_n = {σ ∈ S_n : sgn(σ) = 1}
- Size: |A_n| = n!/2 for n ≥ 2
- Properties: Normal subgroup of S_n with index 2
- Simple groups: A_n is simple for n ≥ 5
- Applications: Fundamental in classification of finite simple groups
Use cycle decomposition to check parity:
- Write in cycle notation: Express as product of disjoint cycles
- Count transpositions: k-cycle = k-1 transpositions
- Sum the counts: Add (length-1) for each cycle
- Check result: Even sum = even permutation
Example: (1 3 5)(2 4) has cycles of length 3 and 2
Transpositions: (3-1) + (2-1) = 2 + 1 = 3 (odd) → odd permutation
The identity permutation requires zero transpositions:
- Definition: Identity leaves every element in place
- Transposition count: 0 transpositions needed
- Parity: 0 is even, so identity is even
- Group theory: Identity must be in every subgroup
- Notation: Written as (1)(2)...(n) or simply e
Permutation parity directly relates to matrix determinants:
- Determinant formula: det(A) = Σ sgn(σ) × a₁σ(1) × ... × aₙσ(n)
- Sign function: sgn(σ) = +1 for even, -1 for odd
- Even permutations: Contribute positively to determinant
- Odd permutations: Contribute negatively to determinant
- Applications: Volume calculations, linear transformations
Related Calculators
Find Calculator
Popular Calculators
Other Calculators
