HEXAGONS , CONICS , A 5 AND PSL 2 ( K ) .sx R. H. DYE .sx 1 .sx Introduction .sx 1.1 When a group occurs geometrically there is a natural expectation that the geometry should account for its subgroups .sx Perhaps this is particularly so for the groups PSL 2 ( K ) and PSL 3 ( K ) that arise from those simplest of geometric objects , the projective line PG ( 1 , K ) and the projective plane PG(2 , K ) over a field K. Yet a simple graphic geometrical raison d' e-circ tre for the existence , when it occurs , of the alternating group A 5 as a subgroup of PSL 2 ( K ) does not seem to be known .sx The initial aim of this paper is to discover a picturesque object that accounts for this occurrence of A 5 .sx We shall show that if PG ( 1 , K ) is represented as a conic C in PG(2 , K ) , then the pertinent object is a certain type of hexagon , distinguished by special concurrencies of its edges .sx For us , a hexagon is a set of six points , no three collinear , in PG(2 , K) :sx we call the six points the vertices of the hexagon , and their 15 joins in pairs its edges .sx And there is a natural concurrent aim :sx obtain the action of PSL 2 ( K ) on these hexagons , and determine their geometry in relation to C. There is , since A 5 is a subgroup of A 6 , a consequent aim :sx discover a configuration of these hexagons that accounts , when it occurs , for A 6 as a subgroup of PSL 3 ( K) .sx Such a configuration has been found , but to keep this paper within reasonable bounds the details will form a sequel .sx However , to paint the whole picture we give a brief description in Section 1.6. 1.2 It is convenient to pause and recall some facts about C. Its group PO 3 ( K ) is a copy of PGL 2 ( K ) , and the commutator subgroup formula is , except when K is the field of two elements , PSL 2 ( K) .sx Now P OMEGA 3 ( K ) is transitive on the points of C and on its chords .sx If K does not have characteristic 2 then C has an associated non-degenerate polarity .sx In that case the poles of the chords are external points , each lying on two tangents .sx There may also be internal points lying on no tangents ; these internal points may not be in a single orbit under PGL 2 ( K ) , that is , may not all be of the same type .sx The polars of internal points are non-secant lines of C. When K is GF( q ) , the finite field with q ( odd ) elements , C has q + 1 points .sx So there are q( q + 1)/2 chords and external points .sx As PG(2 , q ) has formula points there are formula internal points , in this case all of the same type and in a single orbit under formula .sx 1.3 The subgroups of PSL 2 ( q ) have been known since the turn of the century [ 14 ; 12 ; 3 , pp .sx 260 et seq ] :sx Wagner [ 13 , p. 397] gives a convenient non-overlapping list with information about PG(1 , q) .sx This translates readily to C. Some of the subgroups are immediately transparent :sx dihedral groups , cyclic groups , or semidirect products of elementary abelian groups by cyclic groups , that fix chords , non-secant lines or tangents ; the groups of the sub-conics C r ( isomorphic to PG(1 , r) ) that C contains whenever GF( r ) is a subfield of GF( q) .sx Other subgroups occur only when q is odd .sx Then A 4 appears when GF( q ) has characteristic greater than 3 , and SIGMA 4 when , in addition , formula .sx Their existence is not immediate from the geometry of PG(1 , q ) ; their orbit sizes depend on further congruence properties of q [ 13 , p. 397] .sx They are immediately apparent in the richer geometry of C :sx they are the stabilizers of self-polar triangles [ 6 , pp .sx 364 , 369 , 370] .sx The only other subgroup remaining in Wagner's list is A 5 .sx The desire to complete the story and have C account for A 5 is thus pressing .sx Altogether A 5 occurs in three places in the list if q is odd :sx ( i ) if formula and GF( q ) does not have characteristic 3 it occurs and has orbits of lengths 12 , 20 or 30 according to whether formulae respectively ; .sx ( ii ) if GF(9 ) unch GF( q ) it occurs acting transitively on the 10 points of a sub - conic C 9 ; .sx ( iii ) if GF(5 ) unch GF( q ) then A 5 occurs as the PSL 2 ( 5 ) of the sub-conic C 5 , and if formula then formula is also a subgroup of PSL 2 ( q) .sx The hexagons that we shall display account for all these three , apparently dissimilar , appearances of A 5 as a subgroup of PSL 2 ( q) .sx And our hexagons are not restricted to the finite case :sx they exist and account for the occurrence of A 5 in certain PO 3 ( K ) for any field K of characteristic not 2 in which 5 is a square .sx Notice that if 5 is not a square in K then PGL 2 ( K ) contains no element of order 5 so we cannot have A 5 as a subgroup of PSL 2 ( K) :sx a swift direct matrix proof is given in Section 2.7. That A 5 is a subgroup of PSL 2 ( C ) has been known at least since Klein [ 9] found all the finite subgroups of PSL 2 ( C) .sx So we complete the story here too ; all the other finite subgroups of PSL 2 ( C ) are analogues of , and have corresponding interpretations to , other subgroups of PSL 2 ( q) .sx If GF(4 ) is a subfield of K then A 5 occurs in PSL 2 ( K ) as the group PSL 2 ( 4 ) = PGL 2 ( 4 ) fixing the subconic C 4 .sx Although , because of the degeneracy of the polarity of C , this case is not captured by our attack , we show in Section 1.7 that there is a parallel , though distinct , hexagonal explanation .sx 1.4 The main clue to the discovery of the hexagons relevant to A 5 was some recent geometry for C in the case when K is GF(9) .sx It was shown in [ 5] that PO 3 ( 9 ) acts on a set of 12 hexagons defined by the requirement :sx each vertex is an internal point and three edges pass through each of the 10 points on C [ 5 , p. 440] .sx The geometry shows that under P OMEGA 3 ( 9 ) these hexagons form two orbits of size 6 , and that P OMEGA 3 ( 9 ) is A 6 acting , inequivalently , on each orbit [ 5 , p. 443] .sx One then deduces that the stabilizer of such a hexagon in both PO 3 ( 9 ) and P OMEGA 3 ( 9 ) is A 5 :sx it acts on five self-polar triangles .sx There are two reinforcing clues .sx When K is GF(5 ) then P OMEGA 3 ( 5 ) is A 5 , and C has six points which form a hexagon .sx Through each of the 10 internal points passes three chords ; these are just the edges of the hexagon .sx If K is GF(11 ) then Edge [ 6 , p. 380] has presented a hexagon whose vertices are external points , and for which there are 10 internal points at which three edges concur ; its stabilizer in P OMEGA 3 ( 11 ) is A 5 .sx A non-vertex point through which pass three edges of a hexagon has been called a Brianchon point [ 8 , pp .sx 393 et seq .sx ] .sx Since Clebsch encountered [ 2 , p. 336] such a hexagon in PG(2 , R ) when considering the plane representation of the diagonal cubic surface , we shall call a hexagon with ( exactly ) 10 Brianchon points a Clebsch hexagon .sx The simplest example is a regular pentagon and its centre considered as in PG(2 , R ) ; the clues provide three other examples .sx We show first ( Theorem 1 ) that PG(2 , K ) has Clebsch hexagons if and only if K is not of characteristic 2 and 5 is a square in K. Suppose that we have such a K. Which Clebsch hexagons are significant for C ?sx It is not enough to demand that all the vertices are the same type of point , and that so are all the Brianchon points :sx a Clebsch hexagon in PG(2 , C ) has all its vertices and Brianchon points external to any conic not through them ; if C is any conic not through the vertices or Brianchon points of a Clebsch hexagon in PG(2 , q ) then , considered as in PG(2 , q 2 ) , these 16 points are all external to the conic ( containing ) C. For a Clebsch hexagon H in PG(2 , K ) there are ( Theorem 2 ) five triangles each with the property that its three sides contain the six vertices , two on each side , of H. We call these the triangles of H. Motivated by the clues we say that H is a Clebsch hexagon of a conic C if its five triangles are self-polar for C. We show ( Theorem 2 ) that there is a unique orthogonal polarity P for which the triangles of a Clebsch hexagon H are self-polar , and that P corresponds to a conic ; that is , has self-polar points over K , unless K has characteristic 0 and the quadratic form formula is anisotropic .sx It follows ( Theorem 4 ) that , apart from the exceptional case , a conic has Clebsch hexagons .sx The stabilizer of H in both PGL 3 ( K ) and the group of P is A 5 , except when K has characteristic 5 when it is SIGMA 5 ( Theorem 5) .sx Thus , apart from the exceptional case , the stabilizers of the Clebsch hexagons of a conic provide the subgroups A 5 or SIGMA 5 of PGL 2 ( K ) ; and their stabilizers in PSL 2 ( K ) are A 5 or SIGMA 5 , the latter if and only if GF(5 2 ) is a subfield of K. Nothing better can be hoped for .sx In the exceptional case PSL 2 ( K ) does not contain A 5 ( Theorem 5) .sx 1.5 The A 5 of a Clebsch hexagon H of a conic C is ( Section 2.8 ) transitive on the vertices , Brianchon points and edges of H. It is their nature that accounts for the different possibilities for orbits on C described in Section 1.3. The six vertices ( 10 Brianchon points ) lie on a conic if and only if K has characteristic 5 ( 3 ) , when this conic is C :sx there is then an orbit of size 6 ( 10) .sx Otherwise ( Theorem 6 ) , provided they are chords and not non-secants , the 15 edges , 10 polars of the Brianchon points , and six polars of the vertices intersect C in orbits of 30 , 20 , 12 points respectively .sx The group PGL 2 ( K ) is transitive on the Clebsch hexagons of C ; if K is GF( q ) then they form ( Theorem 7 ) two orbits under PSL 2 ( q ) unless the characteristic of K is 5 and GF(5 2 ) is a subfield of K. And , except in characteristic 5 , each triangle of H is shared with one other Clebsch hexagon of C. This sharing gives rise over those GF( q ) with formula or formula to a bipartite 5-valent graph on formula vertices on which PGL 2 ( q ) acts .sx If formula or formula each orbit under PSL 2 ( q ) is a 5-valent graph with formula vertices on which PSL 2 ( q ) acts ( Theorem 7) .sx The smallest non-trivial case occurs for q=31 when there are 248 vertices .sx One other feature deserves highlighting for characteristic not 5 .sx We say , in analogy with the classical case , that two conics have double contact if over K or a quadratic extension of K they touch at two points ; equivalently they define a pencil containing a repeated non-tangent line .sx The six conics through five of the vertices of H have double contact with C. Further , if M is a vertex of H , then all the other vertices of all the Clebsch hexagons of C with M for one vertex lie on the same conic , and each of its points , apart from any contact point with C , is the vertex of one Clebsch hexagon of C ( Theorem 8) .sx 1.6 What of the geometry of the Clebsch hexagons , when they occur , in the plane ?sx They are all of the same type ; PGL 3 ( K ) acts transitively on them ( Theorem 1) .sx Any two triangles of a Clebsch hexagon H are ( Theorem 9 ) in triple perspective from three collinear centres , which are vertices , one from each , of the other three triangles .sx ( There is an additional centre for characteristic 5 .sx ) This is a consequence of there being six lines each containing five vertices of the triangles of H and 10 lines containing three such vertices .sx The full figure of these lines together with the vertices of H , the Brianchon points , the vertices of the triangles and the edges makes a Clebsch hexagon a self-dual configuration ( Theorem 9) .sx And there is more to be told .sx Clebsch hexagons also account for the existence , for certain K , of A 6 as a subgroup of PSL 3 ( K) .sx For reasons of length the details must await a sequel to this paper , but it is pertinent to whet the appetite and to indicate this part of the full story .sx