There are many approaches to constructing the Mathieu

group M_{24}. The exercise below sketches an approach that

may or may not be new.

**Exercise:**

It is well-known that …

There are 56 triangles in an 8-set.

There are 56 spreads in PG(3,2).

The alternating group A_{n }is generated by 3-cycles.

The alternating group A_{8 }is isomorphic to GL(4,2).

Use the above facts, along with the correspondence

described below, to construct M_{24}.

Some background —

A Log24 post of May 19, 2013, cites …

Peter J. Cameron in a 1976 Cambridge U. Press

book — *Parallelisms of Complete Designs* .

See the proof of Theorem 3A.13 on pp. 59 and 60.

See also a Google search for "56 triangles" "56 spreads" Mathieu.

**Update of October 31, 2019 **— A related illustration —

Update of November 2, 2019 —

See also p. 284 of *Geometry and Combinatorics:
Selected Works of J. J. Seidel* (Academic Press, 1991).

That page is from a paper published in 1970.

Update of December 20, 2019 —