Modeling the 21point plane
with outer automorphisms of
S_{6}
by Steven H. Cullinane
The following is adapted from a research note
sent to various mathematicians on December 4, 1986.
The symmetric group S_{n} has no outer automorphisms unless n = 6.
The exceptional behavior of S_{6} 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 2sets and 1sets in a 6set 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 7row by 21column 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 2subset
or 1subset of S, pictured as 2 asterisks, or 1, in a 3x2 box.
An automorphism of S_{n} is determined by its effect on 2cycles, and
the outer automorphisms of S_{6} take each 2cycle to a product of 3
disjoint 2cycles. Our basic trick will be to regard 2cycles as 2sets, and
products of 3 disjoint 2cycles as partitions of S into 3 disjoint 2sets.
The following discussion refers to sections of T shown in figures 1 and 2
below.
Let t map the 15 2cycles L_{i} to the 15 involutions B_{i}
that are products of 3 disjoint 2cycles. Fill in the tablesections L and B
accordingly. Let section A_{i} contain the two 1subsets of S that
appear in the 2cycle (or 2set) L_{i}. Let section P' contain the six
1subsets of S. Let section C_{j} contain the six 2subsets of S that
each contain the 1set 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 1
Sections of the PG(2,4) table in figure
3
L (lines) 
L' 
P (points) 
P' 
A 
C 
B 
FIGURE 2
Subsections of the PG(2,4)in figure 3. The
shaded areas remain constant regardless of which automorphism of S_{6}
is used to build the table.
L_{1} 













L_{15} 
L_{1}' 




L_{6}' 
P_{1} 













P_{15} 
P_{1}' 




P_{6}' 
A_{1} 













A_{15} 
C_{1} 




C_{6} 
B_{1} 













B_{15} 
FIGURE 3
A model of PG(2,4) related to the Mathieu
group M_{24}.
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Page last modified February 5, 2001.
Page created January 24, 2001.