The sign of a permutation is well-defined

The sym­met­ric group \(S_n\) con­tains el­e­ments which are made up from trans­po­si­tions (proof). It is a fact that if \(\sigma \in S_n\) can be made by mul­ti­ply­ing to­gether an even num­ber of trans­po­si­tions, then it can­not be made by mul­ti­ply­ing an odd num­ber of trans­po­si­tions, and vice versa.

knows-req­ui­site(Cyclic group): Equiv­a­lently, there is a group ho­mo­mor­phism from \(S_n\) to the cyclic group \(C_2 = \{0,1\}\), tak­ing the value \(0\) on those per­mu­ta­tions which are made from an even num­ber of per­mu­ta­tions and \(1\) on those which are made from an odd num­ber.

Proof

Let \(c(\sigma)\) be the num­ber of cy­cles in the dis­joint cy­cle de­com­po­si­tion of \(\sigma \in S_n\), in­clud­ing sin­gle­tons. For ex­am­ple, \(c\) ap­plied to the iden­tity yields \(n\), while \(c((12)) = n-1\) be­cause \((12)\) is short­hand for \((12)(3)(4)\dots(n-1)(n)\). We claim that mul­ti­ply­ing \(\sigma\) by a trans­po­si­tion ei­ther in­creases \(c(\sigma)\) by \(1\), or de­creases it by \(1\).

In­deed, let \(\tau = (kl)\). Either \(k, l\) lie in the same cy­cle in \(\sigma\), or they lie in differ­ent ones.

  • If they lie in the same cy­cle, then

    $$\sigma = \alpha (k a_1 a_2 \dots a_r l a_s \dots a_t) \beta$$
    where \(\alpha, \beta\) are dis­joint from the cen­tral cy­cle (and hence com­mute with \((kl)\)). Then \(\sigma (kl) = \alpha (k a_s \dots a_t)(l a_1 \dots a_r) \beta\), so we have bro­ken one cy­cle into two.

  • If they lie in differ­ent cy­cles, then

    $$\sigma = \alpha (k a_1 a_2 \dots a_r)(l b_1 \dots b_s) \beta$$
    where again \(\alpha, \beta\) are dis­joint from \((kl)\). Then \(\sigma (kl) = \alpha (k b_1 b_2 \dots b_s l a_1 \dots a_r) \beta\), so we have joined two cy­cles into one.

There­fore \(c\) takes even val­ues if there are evenly many trans­po­si­tions in \(\sigma\), and odd val­ues if there are odd-many trans­po­si­tions in \(\sigma\).

More for­mally, let \(\sigma = \alpha_1 \dots \alpha_a = \beta_1 \dots \beta_b\), where \(\alpha_i, \beta_j\) are trans­po­si­tions.

knows-req­ui­site(mod­u­lar ar­ith­metic): (The fol­low­ing para­graph is more suc­cinctly ex­pressed as: ”\(c(\sigma) \equiv n+a \pmod{2}\) and also \(\equiv n+b \pmod{2}\), so \(a \equiv b \pmod{2}\).”)
Then \(c(\sigma)\) flips odd-to-even or even-to-odd for each in­te­ger \(1, 2, \dots, a\); it also flips odd-to-even or even-to-odd for each in­te­ger \(1, 2, \dots, b\). There­fore \(a\) and \(b\) must be of the same par­ity.

Parents:

  • Symmetric group

    The sym­met­ric groups form the fun­da­men­tal link be­tween group the­ory and the no­tion of sym­me­try.