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Even Permutation — Definition, Formula & Examples

An even permutation is a permutation that can be written as a product of an even number of transpositions (swaps of two elements). Every permutation is either even or odd, and this classification is called the parity of the permutation.

A permutation σ∈Sn\sigma \in S_n is called even if it can be expressed as a composition of 2k2k transpositions for some non-negative integer kk. Equivalently, σ\sigma is even if and only if sgn⁡(σ)=+1\operatorname{sgn}(\sigma) = +1, where the sign function sgn⁡:Sn→{+1,−1}\operatorname{sgn}: S_n \to \{+1, -1\} is the unique group homomorphism mapping every transposition to −1-1.

Key Formula

sgn⁡(σ)=(−1)N(σ)\operatorname{sgn}(\sigma) = (-1)^{N(\sigma)}
Where:
  • σ\sigma = A permutation in the symmetric group $S_n$
  • N(σ)N(\sigma) = The number of transpositions in any decomposition of $\sigma$
  • sgn⁡(σ)\operatorname{sgn}(\sigma) = The sign (parity) of the permutation; $+1$ if even, $-1$ if odd

How It Works

To determine whether a permutation is even, decompose it into transpositions and count them. If the count is even (including zero), the permutation is even. While the decomposition into transpositions is not unique, the parity of the number of transpositions is always the same for a given permutation — this is a theorem, not obvious. An equivalent method is to write the permutation in cycle notation: a kk-cycle requires exactly k−1k - 1 transpositions, so a kk-cycle is even when kk is odd. Sum up k−1k-1 for each cycle; if the total is even, the permutation is even.

Worked Example

Problem: Determine whether the permutation σ=(1 3 4)(2 5)\sigma = (1\ 3\ 4)(2\ 5) in S5S_5 is even or odd.
Step 1: Count the transpositions needed for each cycle. A kk-cycle decomposes into k−1k - 1 transpositions.
(1 3 4) is a 3-cycle⇒3−1=2 transpositions(1\ 3\ 4) \text{ is a 3-cycle} \Rightarrow 3 - 1 = 2 \text{ transpositions}
Step 2: The second factor is already a transposition (a 2-cycle).
(2 5) is a 2-cycle⇒2−1=1 transposition(2\ 5) \text{ is a 2-cycle} \Rightarrow 2 - 1 = 1 \text{ transposition}
Step 3: Add the transposition counts and check parity.
N(σ)=2+1=3⇒sgn⁡(σ)=(−1)3=−1N(\sigma) = 2 + 1 = 3 \quad \Rightarrow \quad \operatorname{sgn}(\sigma) = (-1)^3 = -1
Answer: The total number of transpositions is 3 (odd), so σ\sigma is an odd permutation, not an even one.

Why It Matters

The set of all even permutations in SnS_n forms the alternating group AnA_n, a normal subgroup of index 2 that plays a central role in group theory. The sign of a permutation also appears in the Leibniz formula for determinants, where each term's sign depends on the parity of the corresponding permutation.

Common Mistakes

Mistake: Confusing cycle length with transposition count. Students sometimes think a 3-cycle needs 3 transpositions.
Correction: A kk-cycle decomposes into k−1k - 1 transpositions. For example, (1 3 4)=(1 4)(1 3)(1\ 3\ 4) = (1\ 4)(1\ 3), which is 2 transpositions, not 3.