Today I Learned

Some of the things I've learned every day since Oct 10, 2016

108: Generating Set for the Symmetric Group

There are many possible choices of a generating set for S_n, the symmetric group. However, one particularly simple one is the set

\{(0, 1, 2, \dots, n-1), (0, 1)\}

where 0, 1, 2, \dots, n-1 is an ordering of the elements being permutated by the elements of S_n, (0, 1, 2, \dots, n-1) is the permutation ‘shifting’ all these elements to their successor, and (0, 1) 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 n elements is a combination of these 2 permutations, because alternating between them in succession allows you to move any of the n elements to an arbitrary position in the ordering.


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