Log24

Info on M13

“Conway describes the beautiful construction of a discrete mathematical structure which he calls ‘M₁₃.’  This structure is a set of 1,235,520 permutations of 13 letters. It is not a group. However, this structure represents the answer to the following group theoretic question:

Why do the simple groups M₁₂ and L₃(3) share some subgroup structure?

In fact, both the Mathieu group M₁₂ and the automorphism group L₃(3) of the projective plane PG(2,3) over GF(3) can be found as subsets of M₁₃.  In addition, M₁₃ is 6-fold transitive, in the sense that it contains enough permutations to map any two 6-tuples made from the thirteen letters into each other.  In this sense, M₁₃ could pass as a parent for both M₁₂ and L₃(3).  As it is known from the classification of primitive groups that there is no finite group which qualifies as a parent in this sense.  Yet, M₁₃ comes close to being a group.

To understand the definition of M₁₃ let us have a look at the projective geometry PG(2,3)….

The points and the lines and the “is-contained-in” relation form an incidence structure over PG(2,3)….

…the 26 objects of the incidence structure [are] 13 points and 13 lines.”

Conway’s construction involves the arrangement, in a circular Levi graph, of 26 marks representing these points and lines, and chords representing the “contains/is contained in” relation.  The resulting diagram has a pleasingly symmetric appearance.

For further information on the geometry of the number 26, one can look up all primitive permutation groups of degree 26.  Conway’s work suggests we look at sets (not just groups) of permutations on n elements.  He has shown that this is a fruitful approach for n=13.  Whether it may also be fruitful for n=26, I do not know.