Alternating group

The al­ter­nat­ing group \(A_n\) is defined as a cer­tain sub­group of the sym­met­ric group \(S_n\): namely, the col­lec­tion of all el­e­ments which can be made by mul­ti­ply­ing to­gether an even num­ber of trans­po­si­tions. This is a well-defined no­tion (proof).

knows-req­ui­site(Nor­mal sub­group): \(A_n\) is a nor­mal sub­group of \(S_n\); it is the quo­tient of \(S_n\) by the sign ho­mo­mor­phism.

Examples

  • A cy­cle of even length is an odd per­mu­ta­tion in the sense that it can only be made by mul­ti­ply­ing an odd num­ber of trans­po­si­tions. For ex­am­ple, \((132)\) is equal to \((13)(23)\).

  • A cy­cle of odd length is an even per­mu­ta­tion, in that it can only be made by mul­ti­ply­ing an even num­ber of trans­po­si­tions. For ex­am­ple, \((1354)\) is equal to \((54)(34)(14)\).

  • The al­ter­nat­ing group \(A_4\) con­sists pre­cisely of twelve el­e­ments: the iden­tity, \((12)(34)\), \((13)(24)\), \((14)(23)\), \((123)\), \((124)\), \((134)\), \((234)\), \((132)\), \((143)\), \((142)\), \((243)\).

Properties

knows-req­ui­site(Nor­mal sub­group): The al­ter­nat­ing group \(A_n\) is of in­dex \(2\) in \(S_n\). There­fore \(A_n\) is nor­mal in \(S_n\) (proof). Alter­na­tively we may give the ho­mo­mor­phism ex­plic­itly of which \(A_n\) is the ker­nel: it is the sign ho­mo­mor­phism.

Children:

Parents:

  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.