Some of the things I've learned every day since Oct 10, 2016
108: Generating Set for the Symmetric Group
February 3, 2017Posted by on
There are many possible choices of a generating set for , the symmetric group. However, one particularly simple one is the set
where is an ordering of the elements being permutated by the elements of , is the permutation ‘shifting’ all these elements to their successor, and is the permutation swapping the first 2 of these.
I won’t provide a formal proof here as it’s a little tedious, but you can easily convince yourself that every permutation of these elements is a combination of these 2 permutations, because alternating between them in succession allows you to move any of the elements to an arbitrary position in the ordering.