A footnote was added to Finite Relativity—

**Background:**

Weyl on what he calls *the relativity problem*—

“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”

– Hermann Weyl, 1949, “Relativity Theory as a Stimulus in Mathematical Research“

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

– Hermann Weyl, 1946, *The Classical Groups *, Princeton University Press, p. 16

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, *M*_{ 24} (containing the original group), acts on the larger array. There is no obvious solution to Weyl’s relativity problem for *M*_{ 24}. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or *symbol-strings* ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is *M*_{ 24}. ….

**Footnote of Sept. 20, 2011:**

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction “The generators of *M*_{ 24} are defined… as permutations of twenty-four 7-cycles in the action of PSL_{2}(7) on seven letters….”

See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis, *Mathematical Proceedings of the Cambridge Philosophical Society* (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, “Natural Constructions of the Mathieu groups,” *Math. Proc. Cambridge Philos. Soc. * (1989), Vol. 106, pp. 423-429

R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups *M*_{ 12} and *M*_{ 24}” In *Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’* (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in *The Atlas of Finite Groups: Ten Years On* , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57