Burnside's lemma

Right at the merge between chemistry and mathematics lies Burnside's lemma, group theory at its best.

Alright, Ambrose Burnside did not invent sideburns and Burnside's lemma. In fact not even William Burnside that it is named after, came up with it. Cauchy and Frobenius knew about it before. But I don't think they are making a fuss about it. Maybe because they are famous anyway [1].

Burnside's lemma can be used for determining isomeric structures. It is a way to systematically find the number of isomeric structures or possible different substitutions. It is a rather laborious way of determining isomeres but it helps you make sure you are not missing anything.

Today I am trying to show what's happening. But you don't really need to know that to apply Burnside's lemma to a problem. Later I am thinking about showing two examples: dichlorobenzenes and dichloro-n-hexanes.

First you need a molecule (let's say benzene) and a substition rule (dichlorobenzenes). Now we look at the molecule fixed in space and make all possible substitutions. This set we call X. It has 6c2 = 15 elements. .

We are not actually interested in this set X. What we would like to know is how many "different" structures there are, meaning structures that cannot be transformed into each other.

To define "different" we need a group G of operations g (bijective functions that act on X). This group G is like the symmetry group, but it makes sense to modify it. First you shouldn't include reflections in order to keep chiral structures. Second you have to include rotations over σ-bonds (unless your are interested in conformeres). [2]

No we can say that two structures in X are different if there is no operation in G that transforms one into the other. X is divided into disjoint subsets of structures that can be transformed into each other by operations in G but only inside the subset. These subsets are called "orbits" [3]. Each one of these orbits corresponds to one isomeric structure (in the example we have ortho, meta, and para orbits each consisting of 5 fixed stuctures).

In other words: We have a set X and a group G acting on it. We are interested in the number of orbits |X/G|. Burnside's lemma states that this number can be found using "fixed sets" X^g. A set X^g consists of all elements in X that stay the same when g is applied (e.g. all the para structures would be the fixed set of the 180° rotation). The proof is not obvious [4] but apparently the following is true:


What we have to do is look at all operations g in G and count the number of elements of X that stay the same when g is applied. From this we know the number of isomeric structures.

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[1] Those days were cool when one single person could come up with hundreds of different new things.

[2] For picky readers: Originally you define the operations as something that is being done to the unsubstituted molecule. The operations in G are X->X (between the fixed structures) but that's just a group isomorphism I think.

[3] Readers that want to keep their chemical maths up can think about how orbits and irreducible representations are related. I think that every irreducible representation is also an orbit.

[4] In other words: I don't know. Or I am just scared of weird terms and symbols (which people are too often I think). Anyway, as long as I can put my problem into mathematical terms I can just ask any mathematician to solve it for me.