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# Modeling the 21-point planewith outer automorphisms of S6

## by Steven H. Cullinane

The following is adapted from a research note sent to various mathematicians on December 4, 1986.

The symmetric group Sn 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 Sn 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 Li to the 15 involutions Bi that are products of 3 disjoint 2-cycles. Fill in the table-sections L and B accordingly. Let section Ai contain the two 1-subsets of S that appear in the 2-cycle (or 2-set) Li. Let section P' contain the six 1-subsets of S. Let section Cj 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.

 L (lines) L' P (points) P' A C B

 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.
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