Modeling the 21-point plane
with outer automorphisms of S₆

by Steven H. Cullinane

The symmetric group S_n has no outer automorphisms unless n = 6. The exceptional behavior of S₆ may be studied via the internal geometry of each of this group's 720 = 6! outer automorphisms. In particular, each such automorphism t yields a model of the projective plane PG(2,4) in which the 21 2-sets and 1-sets in a 6-set S play double roles -- as points, and as lines.

This note merely gives a recipe for such models, without proof that the recipe works. Some evidence, however, is given in an example.

A model of PG(2,4) is specified by giving the 21 points and the 21 lines, and giving a rule that yields the 5 points on each line (or, dually, the 5 lines through each point). Our specifications will describe, given the automorphism t, how to build a 7-row by 21-column table T for a PG(2,4). Row 1 will be a list of lines, row 2 a list of points. The last 5 entries in each column will give the 5 points on the line heading that column, and at the same time give the 5 lines through the point in entry 2 of that column. Each entry in T will be a 2-subset or 1-subset of S, pictured as 2 asterisks, or 1, in a 3x2 box.

An automorphism of S_n is determined by its effect on 2-cycles, and the outer automorphisms of S₆ take each 2-cycle to a product of 3 disjoint 2-cycles. Our basic trick will be to regard 2-cycles as 2-sets, and products of 3 disjoint 2-cycles as partitions of S into 3 disjoint 2-sets.

The following discussion refers to sections of T shown in figures 1 and 2 below.

Let t map the 15 2-cycles L_i to the 15 involutions B_i that are products of 3 disjoint 2-cycles. Fill in the table-sections L and B accordingly. Let section A_i contain the two 1-subsets of S that appear in the 2-cycle (or 2-set) L_i. Let section P' contain the six 1-subsets of S. Let section C_j contain the six 2-subsets of S that each contain the 1-set P'_j.

If the table thus far can be completed to a PG(2,4) -- which has not been proved, since this recipe says "how," not "why" -- it is clear that there will be only one way to fill out the remaining sections, P and L' (in that order).

For evidence that the recipe does work, see figure 3 below.

Note that in all tables T constructed by this method, only sections P, L', and B vary from table to table, depending on the automorphism t that is used. Note also that if t is of order 2, lines and points become indistinguishable, and only one row (not two) of headings is needed.

FIGURE 2
Subsections of the PG(2,4)in figure 3. The shaded areas remain constant regardless of which automorphism of S₆ is used to build the table.

L1

L15

L1'

L6'

P1

P15

P1'

P6'

A1

A15

C1

C6

B1

B15

FIGURE 3
A model of PG(2,4) related to the Mathieu group M₂₄.

Lines ...... *... *... ...... ...* ...* *... ...... *... ...* ...... ...* *... *... ...... ...* ...* ...... ...... *... ...* ...... ...* *... *... ...... ...* ...* ...... *... *... ...* ...... ...* *... ...... *..* ...... ...... ...... *..* ...... ...... ...... *..* *... ...... ...... ...* ...... ...... ...... *... ...... ...... ...* ...... ...... ...... *... ...... ...... ...* Points ...... *... *... ...... ...* ...* *... *... ...... ...* ...* ...... *... ...... *... ...* ...... ...* ...... ...* *... ...... *... ...* *... ...* ...... ...* *... ...... *... ...... ...* ...* ...... *... *..* ...... ...... ...... ...... *..* ...... *..* ...... *... ...... ...... ...* ...... ...... ...... ...... *... ...... ...... ...* ...... *... ...... ...... ...* ...... Points on above lines (or lines on above points) ...... *... ...... ...... ...* ...... *... ...... ...... ...* ...... ...... *... ...... ...... ...* ...... ...... ...... *... ...... ...... ...* ...... *... ...... ...... ...* ...... ...... *... ...... ...... ...* ...... ...... *... ...... ...... ...... *... ...... ...... ...... *... *... *... ...... ...* ...* ...... ...... *... *... ...... ...* ...* ...... *... *... ...... ...* ...* ...... ...... *... ...... ...... ...* ...... ...... *... ...... ...... ...* ...... *... ...... ...... ...* ...... ...... ...... ...* ...... ...... *... ...... ...... ...* ...... ...... *... ...... ...* ...... ...... *... ...... ...* ...... ...... ...... ...* ...... ...... ...... ...* *... ...... *... ...* ...... ...* *... ...... *... ...* ...... ...* *... *... ...... ...* ...* ...... *..* ...... ...... *..* ...... ...... ...* ...... *... ...* *... ...... ...* *... ...... ...* ...... *... *... *... ...... *... ...... *... *... ...... *... *... *... ...... *... *... ...... *... ...... *... ...... *... *... ...... *... *... ...... *... *... *... ...* ...... ...* *... ...... ...... ...* *... ...... *... ...* ...... *... ...* ...... ...* *... ...... *..* ...... ...... ...* *... *... ...* ...... *... ...* ...... ...... ...* *... ...... *..* ...... ...... ...* ...* ...... ...* ...* ...* ...... ...* ...* ...... ...* ...* ...* ...... ...* ...* ...... ...... ...* ...* ...* ...... ...* ...* ...* ...... *... ...... ...* ...* ...... *... ...* ...... *... *... ...... ...* ...* *... ...... *... ...* ...... ...... ...... *..* ...... *... ...* ...... *... ...* ...... ...... *..* *... ...... ...* *... ...... ...* ...* ...... *... ...* *... ...... ...... *..* ...... ...... ...* *... ...... ...... *..* ...... *... ...* *..* ...... ...... *... ...* ...... *... ...... ...* *..* ...... ...... *..* ...... ...... ...... ...... *..* ...... ...... *..* ...... *..* ...... ...... *..* ......