May 26, 2003

• Mental Health Month, Day 26:

  
  

  Many Dimensions,
  
  Part III — Why 26?

  
  

  At first blush, it seems unlikely that the number 26=2×13, as a product of only two small primes (and those distinct) has any purely mathematical properties of interest. (On the other hand, consider the number 6.)  Parts I and II of “Many Dimensions,” notes written earlier today, deal with the struggles of string theorists to justify their contention that a space of 26 dimensions may have some significance in physics.  Let them struggle.  My question is whether there are any interesting purely mathematical properties of 26, and it turns out, surprisingly, that there are some such properties. All this is a longwinded way of introducing a link to the web page titled “Info on M13,” which gives details of a 1997 paper by J. H. Conway*.

  
  
  
  
  

  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.

  
  

  There is no obvious connection to physics, although the physics writer John Baez quoted in my previous two entries shares Conway’s interest in the Mathieu groups. 

  
  

   * J. H. Conway, “M₁₃,” in Surveys in Combinatorics, 1997, edited by R. A. Bailey, London Mathematical Society Lecture Note Series, 241, Cambridge University Press, Cambridge, 1997. 338 pp. ISBN 0 521 59840 0.
  
  

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