Lagrange theorem on subgroup size

La­grange’s The­o­rem states that if \(G\) is a finite group and \(H\) a sub­group, then the or­der \(|H|\) of \(H\) di­vides the or­der \(|G|\) of \(G\). It gen­er­al­ises to in­finite groups: the state­ment then be­comes that the left cosets form a par­ti­tion, and for any pair of cosets, there is a bi­jec­tion be­tween them.

Proof

In full gen­er­al­ity, the cosets form a par­ti­tion and are all in bi­jec­tion.

To spe­cial­ise this to the finite case, we have di­vided the \(|G|\) el­e­ments of \(G\) into buck­ets of size \(|H|\) (namely, the cosets), so \(|G|/|H|\) must in par­tic­u­lar be an in­te­ger.

Children:

Parents:

  • Group

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