Symmetric group on a finite set is 2-generated

Statement
The fact about::symmetric group on a finite set is a fact about::2-generated group: it can be generated by two elements.

Note that when the finite set has 3 or more elements (so the degree is 3 or more), the symmetric group is not cyclic, so we obtain that the fact about::minimum size of generating set is precisely 2.

Stronger facts

 * Dixon's theorem: This states that two randomly picked elements have a high chance of generating the whole symmetric group.

Also refer presentation theory of symmetric groups to learn about good choices of presentation for symmetric groups.

Facts used

 * 1) Transpositions of adjacent elements generate the symmetric group on a finite set

A transposition and a cycle
Consider the set $$\{ 1,2,3, \dots, n \}$$. We show that the symmetric group on this set is generated by the permutations:

$$(1,2)$$ and $$(1,2,3,\dots,n)$$.

Proof: Observe that, for $$0 \le i < n -1$$:

$$(1,2,3,\dots,n)^i (1,2) (1,2,3,\dots,n)^{-i} = (i+1,i+2)$$

Thus, all transpositions of adjacent elements are in the subgroup generated by these two permutations. Using fact (1), we see that these two permutations generate the whole group.