Log24

From "The Mathieu Group M₁₂ and Conway’s M₁₃-Game" (pdf), senior honors thesis in mathematics by Jeremy L. Martin under the supervision of Professor Noam D. Elkies, Harvard University, April 1, 1996–

"Let P₃ denote the projective plane of order 3. The standard construction of P₃ is to remove the zero point from a three-dimensional vector space over the field F₃ and then identify each point x with -x, obtaining a space with (3³ – 1)/2 = 13 points. However, we will be concerned only with the geometric properties of the projective plane. The 13 points of P₃ are organized into 13 lines, each line containing four points. Every point lies on four lines, any two points lie together on a unique line, and any two lines intersect at a unique point….

Conway [3] proposed the following game…. Place twelve numbered counters on the points… of P₃ and leave the thirteenth point… blank. (The empty point will be referred to throughout as the "hole.") Let the location of the hole be p; then a primitive move of the game consists of selecting one of the lines containing the hole, say {p, q, r, s}. Move the counter on q to p (thus moving the hole to q), then interchange the counters on r and s….

There is an obvious characterization of a move as a permutation in S₁₃, operating on the points of P₃. By limiting our consideration to only those moves which return the hole to its starting point…. we obtain the Conway game group. This group, which we shall denote by G_C, is a subgroup of the symmetric group S₁₂ of permutations of the twelve points…, and the group operation of G_C is concatenation of paths. Conway [3] stated, but did not prove explicitly, that G_C is isomorphic to the Mathieu group M₁₂. We shall subsequently verify this isomorphism.

The set of all moves (including those not fixing the hole) is given the name M₁₃ by Conway. It is important that M₁₃ is not a group…."

[3] John H. Conway, "Graphs and Groups and M₁₃," Notes from New York Graph Theory Day XIV (1987), pp. 18–29.

"Conway’s puzzle M₁₃ involves the 13 points and 13 lines of P₃. On all but one point numbered counters are placed holding the numbers 1,…,12 and a move involves interchanging one counter and the 'hole' (the unique point having no counter) and interchanging the counters on the two other points of the line determined by the first two points. In the picture [above] the lines are represented by dashes around the circle in between two counters and the points lying on this line are those that connect to the dash either via a direct line or directly via the circle. In the first part we saw that the group of all reachable positions in Conway's M₁₃ puzzle having the hole at the top position contains the sporadic simple Mathieu group M₁₂ as a subgroup."