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
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