The collection of even-signed permutations is a group

The col­lec­tion of el­e­ments of the sym­met­ric group \(S_n\) which are made by mul­ti­ply­ing to­gether an even num­ber of per­mu­ta­tions forms a sub­group of \(S_n\).

This proves that the al­ter­nat­ing group \(A_n\) is well-defined, if it is given as “the sub­group of \(S_n\) con­tain­ing pre­cisely that which is made by mul­ti­ply­ing to­gether an even num­ber of trans­po­si­tions”.

Proof

Firstly we must check that “I can only be made by mul­ti­ply­ing to­gether an even num­ber of trans­po­si­tions” is a well-defined no­tion; this is in fact true.

We must check the group ax­ioms.

  • Iden­tity: the iden­tity is sim­ply the product of no trans­po­si­tions, and \(0\) is even.

  • As­so­ci­a­tivity is in­her­ited from \(S_n\).

  • Clo­sure: if we mul­ti­ply to­gether an even num­ber of trans­po­si­tions, and then a fur­ther even num­ber of trans­po­si­tions, we ob­tain an even num­ber of trans­po­si­tions.

  • In­verses: if \(\sigma\) is made of an even num­ber of trans­po­si­tions, say \(\tau_1 \tau_2 \dots \tau_m\), then its in­verse is \(\tau_m \tau_{m-1} \dots \tau_1\), since a trans­po­si­tion is its own in­verse.

Parents:

  • Alternating group

    The al­ter­nat­ing group is the only nor­mal sub­group of the sym­met­ric group (on five or more gen­er­a­tors).