## 305 – Derived subgroups of symmetric groups

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:

The output of the program is as follows:

 [(), (1,3,2), (1,2,3)] 3 True * * * [(), (2,4,3), (2,3,4), (1,4,3), (1,3,4), (1,2)(3,4), (1,3,2), (1,4)(2,3), (1,2,3), (1,4,2), (1,3)(2,4), (1,2,4)] 12 True * * * [(), (3,5,4), (3,4,5), (2,5,4), (2,4,5), (2,3)(4,5), (1,5,4), (1,4,5), (1,3)(4,5), (1,2)(4,5), (2,4,3), (2,5)(3,4), (2,3,4), (1,4,3), (1,5)(3,4), (1,3,4), (1,2)(3,4), (2,5,3), (2,4)(3,5), (1,5,3), (1,4)(3,5), (1,5,3,4,2), (1,4,5,3,2), (1,3,4,5,2), (1,2,5,3,4), (1,2,4,5,3), (1,2,3,4,5), (2,3,5), (1,3,5), (1,2,5,4,3), (1,2,4,3,5), (1,2,3,5,4), (1,5,4,3,2), (1,4,3,5,2), (1,3,5,4,2), (1,2)(3,5), (1,3,2), (1,4)(2,3), (1,5)(2,3), (1,2,3), (1,4,3,2,5), (1,5,3,2,4), (1,4,2,3,5), (1,5,2,3,4), (1,4,2), (1,3)(2,4), (1,3,4,2,5), (1,5,4,2,3), (1,5,2), (1,5,2,4,3), (1,4)(2,5), (1,3)(2,5), (1,3,5,2,4), (1,3,2,4,5), (1,5)(2,4), (1,4,2,5,3), (1,3,2,5,4), (1,4,5,2,3), (1,2,4), (1,2,5)] 60 True * * * [(), (4,6,5), (4,5,6), (3,6,5), (3,5,6), (3,4)(5,6), (2,6,5), (2,5,6), (2,4)(5,6), (2,3)(5,6), (1,6,5), (1,5,6), (1,4)(5,6), (1,3)(5,6), (1,2)(5,6), (3,5,4), (3,6)(4,5), (3,4,5), (2,5,4), (2,6)(4,5), (2,4,5), (2,3)(4,5), (1,5,4), (1,6)(4,5), (1,4,5), (1,3)(4,5), (1,2)(4,5), (3,6,4), (3,5)(4,6), (2,6,4), (2,5)(4,6), (2,6,4,5,3), (2,5,6,4,3), (2,4,5,6,3), (2,3,6,4,5), (2,3,5,6,4), (2,3,4,5,6), (1,6,4), (1,5)(4,6), (1,6,4,5,3), (1,5,6,4,3), (1,4,5,6,3), (1,3,6,4,5), (1,3,5,6,4), (1,3,4,5,6), (1,6,4,5,2), (1,5,6,4,2), (1,4,5,6,2), (1,3,2)(4,5,6), (1,2,6,4,5), (1,2,5,6,4), (1,2,4,5,6), (1,2,3)(4,5,6), (3,4,6), (2,4,6), (2,3,6,5,4), (2,3,5,4,6), (2,3,4,6,5), (2,6,5,4,3), (2,5,4,6,3), (2,4,6,5,3), (1,4,6), (1,3,6,5,4), (1,3,5,4,6), (1,3,4,6,5), (1,6,5,4,3), (1,5,4,6,3), (1,4,6,5,3), (1,2,6,5,4), (1,2,5,4,6), (1,2,4,6,5), (1,2,3)(4,6,5), (1,6,5,4,2), (1,5,4,6,2), (1,4,6,5,2), (1,3,2)(4,6,5), (2,3)(4,6), (1,3)(4,6), (1,2)(4,6), (2,4,3), (2,5)(3,4), (2,6)(3,4), (2,3,4), (1,4,3), (1,5)(3,4), (1,6)(3,4), (1,3,4), (1,2)(3,4), (2,5,4,3,6), (2,6,4,3,5), (2,5,3,4,6), (2,6,3,4,5), (1,5,4,3,6), (1,6,4,3,5), (1,5,3,4,6), (1,6,3,4,5), (1,2)(3,6,5,4), (1,2)(3,5,6,4), (1,2)(3,4,6,5), (1,2)(3,4,5,6), (1,5,2,6)(3,4), (1,6,2,5)(3,4), (1,4,3)(2,6,5), (1,4,3)(2,5,6), (1,6,5)(2,4,3), (1,5,6)(2,4,3), (1,3,4)(2,6,5), (1,3,4)(2,5,6), (1,3,2,4)(5,6), (1,6,5)(2,3,4), (1,5,6)(2,3,4), (1,4,2,3)(5,6), (2,5,3), (2,4)(3,5), (2,4,5,3,6), (2,6,5,3,4), (1,5,3), (1,4)(3,5), (1,4,5,3,6), (1,6,5,3,4), (1,5,3,4,2), (1,6,2)(3,4,5), (1,4,5,3,2), (1,3,4,5,2), (1,2,5,3,4), (1,2,6)(3,4,5), (1,2,4,5,3), (1,2,3,4,5), (2,6,3), (2,6,3,5,4), (2,5)(3,6), (2,4)(3,6), (2,4,6,3,5), (2,4,3,5,6), (1,6,3), (1,6,3,5,4), (1,5)(3,6), (1,4)(3,6), (1,4,6,3,5), (1,4,3,5,6), (1,6,3,4,2), (1,5,2)(3,4,6), (1,6,3,2)(4,5), (1,5,6,3,2), (1,4,5,2)(3,6), (1,4,2)(3,5,6), (1,5,6,2)(3,4), (1,3,4,2)(5,6), (1,6,3,4)(2,5), (1,5)(2,6,3,4), (1,6,3)(2,4,5), (1,5,6,3)(2,4), (1,4,5)(2,6,3), (1,4)(2,5,6,3), (1,6)(2,3,4,5), (1,4,5,6)(2,3), (1,3,4,5)(2,6), (1,3)(2,4,5,6), (1,2,6,3,4), (1,2,5)(3,4,6), (1,2,6,3)(4,5), (1,2,5,6,3), (1,2,4,5)(3,6), (1,2,4)(3,5,6), (1,2,5,6)(3,4), (1,2,3,4)(5,6), (2,6)(3,5), (2,5,3,6,4), (2,4,3,6,5), (2,5,6,3,4), (1,6)(3,5), (1,5,3,6,4), (1,4,3,6,5), (1,5,6,3,4), (1,4,2)(3,6,5), (1,4,6,2)(3,5), (1,6,5,3,2), (1,5,3,2)(4,6), (1,3,4,6,2), (1,6,5,2)(3,4), (1,2,6,5,3), (1,2,5,3)(4,6), (1,2,4)(3,6,5), (1,2,4,6)(3,5), (1,2,6,5)(3,4), (1,2,3,4,6), (1,6)(2,5,3,4), (1,5,3,4)(2,6), (1,6,5,3)(2,4), (1,5,3)(2,4,6), (1,4)(2,6,5,3), (1,4,6)(2,5,3), (1,5)(2,3,4,6), (1,4,6,5)(2,3), (1,3,4,6)(2,5), (1,3)(2,4,6,5), (1,4,6,3,2), (1,2,4,6,3), (2,3,5), (1,3,5), (1,2,5,4,3), (1,2,6)(3,5,4), (1,2,4,3,5), (1,2,3,5,4), (1,5,4,3,2), (1,6,2)(3,5,4), (1,4,3,5,2), (1,3,5,4,2), (1,2,6,4,3), (1,2,5)(3,6,4), (1,2,4,3)(5,6), (1,2,6,4)(3,5), (1,2,3,5)(4,6), (1,2,3,5,6), (1,5,2)(3,6,4), (1,6,4,3,2), (1,4,3,2)(5,6), (1,3,5,2)(4,6), (1,3,5,6,2), (1,6,4,2)(3,5), (1,5)(2,6,4,3), (1,6,4,3)(2,5), (1,4,3,5)(2,6), (1,6)(2,4,3,5), (1,3,5)(2,6,4), (1,3)(2,5,6,4), (1,3,5,6)(2,4), (1,6,4)(2,3,5), (1,5,6,4)(2,3), (1,4)(2,3,5,6), (1,2)(3,5), (1,6,3,5,2), (1,2,6,3,5), (1,5,3)(2,6,4), (1,6,4)(2,5,3), (1,4,2,6)(3,5), (1,6,2,4)(3,5), (1,3,2,5)(4,6), (1,3,5)(2,4,6), (1,5,2,3)(4,6), (1,4,6)(2,3,5), (1,2)(3,6,4,5), (1,2)(3,5,4,6), (1,6,3)(2,5,4), (1,5,4)(2,6,3), (1,6,3,5)(2,4), (1,5)(2,4,6,3), (1,4)(2,6,3,5), (1,4,6,3)(2,5), (1,6)(2,3,5,4), (1,5,4,6)(2,3), (1,3,5,4)(2,6), (1,3)(2,5,4,6), (1,5,4,2)(3,6), (1,2,5,4)(3,6), (2,3,6), (1,3,6), (1,2,4,3,6), (1,2,3,6)(4,5), (1,2,3,6,5), (1,5,4,3)(2,6), (1,6)(2,5,4,3), (1,4,3,6)(2,5), (1,5)(2,4,3,6), (1,3)(2,6,5,4), (1,3,6)(2,5,4), (1,3,6,5)(2,4), (1,6,5,4)(2,3), (1,5,4)(2,3,6), (1,4)(2,3,6,5), (1,4,3,6,2), (1,3,6,2)(4,5), (1,3,6,5,2), (1,2,3,6,4), (1,3,6,4,2), (1,2,5,3,6), (1,5,3,6,2), (1,2)(3,6), (1,4,5,3)(2,6), (1,4)(2,5,3,6), (1,6)(2,4,5,3), (1,5,3,6)(2,4), (1,3)(2,6,4,5), (1,3,6,4)(2,5), (1,3,6)(2,4,5), (1,6,4,5)(2,3), (1,5)(2,3,6,4), (1,4,5)(2,3,6), (1,5,2,4)(3,6), (1,4,2,5)(3,6), (1,6,2,3)(4,5), (1,3,2,6)(4,5), (1,3,2), (1,4)(2,3), (1,5)(2,3), (1,6)(2,3), (1,2,3), (1,5,3,2,6), (1,6,3,2,5), (1,5,2,3,6), (1,6,2,3,5), (1,4,3,2,5), (1,5,3,2,4), (1,4,2,3,5), (1,5,2,3,4), (1,4,3,2,6), (1,4,2,3,6), (1,5,2,6,4), (1,6,4,2,5), (1,4,5,2,6), (1,4,2,5,6), (1,6,2,4,5), (1,5,6,2,4), (1,6,3,2,4), (1,3,2,6,5), (1,3,2,5,4), (1,3,2,4,6), (1,6,2,3,4), (1,6,5,2,3), (1,5,4,2,3), (1,4,6,2,3), (1,5,4,2,6), (1,6,2,5,4), (1,4,2,6,5), (1,4,6,2,5), (1,6,5,2,4), (1,5,2,4,6), (1,3,2,6,4), (1,3,2,5,6), (1,3,2,4,5), (1,6,4,2,3), (1,5,6,2,3), (1,4,5,2,3), (1,4,2), (1,3)(2,4), (1,3,4,2,5), (1,3,4,2,6), (1,6)(2,4), (1,5)(2,4), (1,4,2,6,3), (1,4,2,5,3), (1,5,2), (1,5,2,4,3), (1,4)(2,5), (1,3)(2,5), (1,3,5,2,4), (1,6,2), (1,6,2,4,3), (1,5)(2,6), (1,6,2,5,3), (1,4)(2,6), (1,3)(2,6), (1,3,6,2,5), (1,3,6,2,4), (1,6)(2,5), (1,3,5,2,6), (1,5,2,6,3), (1,2,4), (1,2,5), (1,2,6)] 360 True * * *

This program is too slow to verify whether the same holds for $i>6$. Instead, the following modification allows us to verify that $(S_7)'=A_7$, without checking whether every element of $A_7$ is indeed a commutator. The program lists the size of $A_7$, and then the size of the list generated by what is essentially closing the list of commutators under products. A counter keeps track of how many elements this list has, and for no good reason other than to keep track of how fast the program is running (perhaps with some $i$ other than 7), along the way I list the values of the counter that are multiples of 1000.

The output is in this case:

2520
1000
2000
2520