305 – Derived subgroups of symmetric groups

April 11, 2012

One of the problems in the last homework set is to study the derived group of the symmetric group $S_n$.

Recall that if $G$ is a group and $a,b\in G$, then their commutator is defined as

${}[a,b]=aba^{-1}b^{-1}$.

The derived group $G'$ is the subgroup of $G$ generated by the commutators.

Note that, since any permutation has the same parity as its inverse, any commutator in $S_n$ is even. This means that $G'\le A_n$.

The following short program is Sage allows us to verify that, for $3\le i\le 6$, every element of $(S_i)'$ is actually a commutator. The program generates a list of the commutators of $S_i$, then verifies that this list is closed under products and inverses (so it is a group). It also lists the size of this group. Note that the size is precisely ${}|A_i|$, so $(S_i)'=A_i$ in these 4 cases: