Exercise 1709Algebra › GroupsL1Added 3 Nov 2026

The sign homomorphism

#sign#parity#homomorphism

Statement

Show that the sign map $\varepsilon:S_n\to\{1,-1\}$ is a group homomorphism, and compute its kernel.

Hint

$\varepsilon(\sigma)=(-1)^{N(\sigma)}$ where $N(\sigma)$ is the number of inversions. Show $N(\sigma\tau)\equiv N(\sigma)+N(\tau)\pmod 2$ .

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Solution

Homomorphism. For any transposition $\tau$ , the number of inversions of $\sigma\tau$ differs from that of $\sigma$ by an odd number (since swapping two positions changes the parity of the number of inversions). By induction, decomposing a permutation into transpositions shows that the parity of the number of inversions is a homomorphism to $\{1,-1\}$ .

Kernel. $\ker\varepsilon=\{\sigma:\varepsilon(\sigma)=1\}=A_n$ , the alternating group (even permutations).

Order. $|A_n|=n!/2$ for $n\ge2$ (by the first isomorphism theorem, since $\varepsilon$ is surjective).

Conclusion. $\varepsilon$ is a surjective homomorphism with kernel $A_n$ .

The sign homomorphism

Statement

Solution

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