# Symmetric group on a finite set is 2-generated

## Statement

The Symmetric group (?) on a finite set is a 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 Minimum size of generating set (?) is precisely 2.

## Related facts

### 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

## Proof

### 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.