Sign homomorphism (from the symmetric group)

The sign ho­mo­mor­phism is given by send­ing a per­mu­ta­tion \(\sigma\) in the sym­met­ric group \(S_n\) to \(0\) if we can make \(\sigma\) by mul­ti­ply­ing to­gether an even num­ber of trans­po­si­tions, and to \(1\) oth­er­wise.

knows-req­ui­site(mod­u­lar ar­ith­metic): Equiv­a­lently, it is given by send­ing \(\sigma\) to the num­ber of trans­po­si­tions mak­ing it up, mod­ulo \(2\).

The sign ho­mo­mor­phism is well-defined.

knows-req­ui­site(quo­tient group): The al­ter­nat­ing group is ob­tained by tak­ing the quo­tient of the sym­met­ric group by the sign ho­mo­mor­phism.

Parents:

  • Symmetric group

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