Structure of the truncated icosahedron (e.g. fullerene or \(\text{C}_{60}\), viral coatings) and a \(60\)-element conjugacy class in \(\text{PSl}(2,11)\).

*(English)*Zbl 0830.20073The truncated icosahedron is a polyhedron \(P\) with 60 vertices \(V\), 90 edges \(E\), 12 pentagons and 20 hexagons as faces. The icosahedral group \(A\) of the proper symmetry rotations of the icosahedron \(\pi\), which is isomorphic to \(A_5\), is also a symmetry group for \(P\). The action of \(A\) on the graph \(\Delta\) of the vertices \(V\) and edges \(E\) makes \(A\) a symmetry group of \(\Delta\). Furthermore, the vertices \(V\) form a principal homogeneous space for \(A\) and the structure of \(\Delta\) (which determines \(P\)) is given by the pairs of vertices yielding \(E\).

The idea of the author is to investigate by group-theoretical methods the structure of \(\Delta\) by making the elements of \(V\) group elements. Considered is the group \(\text{PSl} (2,11)\) of order 660 which operates transitively on a set of 11 elements (taken to be the finite field \(F_{11})\) and has the icosahedral group as an isotropy subgroup \(A\). Once such a subgroup \(A\) of \(\text{PSl} (2,11)\) is chosen, there is a corresponding conjugacy class \(M\) of elements of order 11 in \(\text{PSl} (2,11)\) which is \(A\)-invariant (under conjugation) and has 60 elements. One can, accordingly, make \(M\) a model of \(V\). The vertices are now group elements of order 11. The edge structure \(E\) can then be expressed in terms of order 2 elements \(x^{-1} y\), with \(x, y \in V\).

In this way, one gets a group theoretical model of \(\Delta\), and thus one of \(P\). In the paper this model is extensively investigated. In particular, the 12 pentagons correspond to the 12 sets of maximally commutative subsets of \(M\). There are many more relations revealing the rich mathematical structure associated with the geometry of a truncated icosahedron.

The idea of the author is to investigate by group-theoretical methods the structure of \(\Delta\) by making the elements of \(V\) group elements. Considered is the group \(\text{PSl} (2,11)\) of order 660 which operates transitively on a set of 11 elements (taken to be the finite field \(F_{11})\) and has the icosahedral group as an isotropy subgroup \(A\). Once such a subgroup \(A\) of \(\text{PSl} (2,11)\) is chosen, there is a corresponding conjugacy class \(M\) of elements of order 11 in \(\text{PSl} (2,11)\) which is \(A\)-invariant (under conjugation) and has 60 elements. One can, accordingly, make \(M\) a model of \(V\). The vertices are now group elements of order 11. The edge structure \(E\) can then be expressed in terms of order 2 elements \(x^{-1} y\), with \(x, y \in V\).

In this way, one gets a group theoretical model of \(\Delta\), and thus one of \(P\). In the paper this model is extensively investigated. In particular, the 12 pentagons correspond to the 12 sets of maximally commutative subsets of \(M\). There are many more relations revealing the rich mathematical structure associated with the geometry of a truncated icosahedron.

Reviewer: A.Janner (Nijmegen)

##### MSC:

20H15 | Other geometric groups, including crystallographic groups |

51F15 | Reflection groups, reflection geometries |

52B15 | Symmetry properties of polytopes |

82D25 | Statistical mechanical studies of crystals |

##### Keywords:

fullerene; truncated icosahedron; icosahedral group; symmetry rotations; action; homogeneous space; isotropy subgroup; conjugacy class; commutative subsets
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\textit{B. Kostant}, Sel. Math., New Ser. 1, No. 1, 163--195 (1995; Zbl 0830.20073)

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