The collection of even-signed permutations is a group

https://arbital.com/p/even_signed_permutations_form_a_group

by Patrick Stevens Jun 17 2016

This proves the well-definedness of one particular definition of the alternating 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.