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:

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305 – Cube moves

April 11, 2012

Here is a small catalogue of moves of the Rubik’s cube. Appropriately combining them and their natural analogues under rotations or reflections, allow us to solve Rubik’s cube starting from any (legal) position. I show the effect the moves have when applied to the solved cube.

But, first, some relevant links:

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