View source for Signature (permutation)

''sign of a permutation''

The sign or signature of a permutation of a finite set, which we can identify with $\{1,2,\ldots,n\}$ for some $n$, is a multiplicative map $\epsilon$ from the group of permutations $S_n$ to $\pm 1$.  Permutations with sign $+1$ are ''even'' and those with sign $-1$ are ''odd''.  The sign may be defined in a number of ways.
A simple formula is
$$
\epsilon(\pi) = \frac{\prod_{1 \le i < j \le n} (x^{\pi(i)}-x^{\pi(j)})}{\prod_{1 \le i < j \le n} (x^i-x^j)} \ .\label{1}
$$

The sign of $\pi$ may also be defined by the parity of the number of [[transposition]]s which compose $\pi$: this is well-defined since an odd number of transpositions cannot give the identity.  A related definition is that the sign is the parity of  the [[decrement]] $n - c$ where $c = c(\pi)$ is the number of cycles (orbits) of $\pi$.   

Since $\epsilon$ is a homomorphism from $S_n$ to $C_2 = \{\pm1\}$, the ''[[alternating group]]'', or group of even permutations, $A_n$ is the kernel of $\epsilon$ and so a [[normal subgroup]] of $S_n$.

The definition of $\epsilon$ may be extended to maps which are not permutations by defining it to be zero.    This is consistent with (1) and also with the use of the [[Kronecker symbol]] in tensor notation. 

====References====
* Walter Ledermann "Introduction to the theory of finite groups" Oliver and Boyd (1957) {{ZBL|0131.25503}}

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