Abstract
For any given finite group, Schulte and Williams (Discrete Comput Geom 54(2):444–458, 2015) establish the existence of a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the convex polytope we build for the given finite group is binary, and even combinatorial in the sense of Naddef and Pulleyblank (J Combin Theory Ser B 31(3):297–312, 1981); the diameter of its skeleton is at most 2; any combinatorial automorphism of the polytope is induced by some isometry of the space; any automorphism of the skeleton is a combinatorial automorphism.
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References
Arlinghaus, W.C.: The Classification of Minimal Graphs with Given Abelian Automorphism Group. Memoirs of the American Mathematical Society, vol. 330. American Mathematical Society, Providence (1985)
Babai, L.: On the minimum order of graphs with given group. Can. Math. Bull. 17(4), 467–470 (1974)
Babai, L.: Symmetry groups of vertex-transitive polytopes. Geom. Dedicata 6(3), 331–337 (1977)
Babai, L.: On the abstract group of automorphisms. In: Temperley, H. (ed.) Combinatorics. London Mathematical Society Lecture Notes, vol. 52, pp. 1–40. Cambridge University Press, Cambridge (1981)
Babai, L., Godsil, C.D.: On the automorphism groups of almost all Cayley graphs. Eur. J. Comb. 3(1), 9–15 (1982)
Billera, L., Sarangarajan, A.: All 0–1 polytopes are traveling salesman polytopes. Combinatorica 16(2), 175–188 (1996)
Frucht, R.: Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compos. Math. 6, 239–250 (1939)
Godsil, C.D.: GRRs for nonsolvable groups. In: Algebraic Methods in Graph Theory, Vol. I, II. Colloquia Mathematica Societatis János Bolyai, vol. 25, pp. 221–239. North-Holland, Amsterdam (1981)
Hevia, H.: Grafos con grupo dado de automorfismos. Proyecciones 14, 139–167 (2011)
Matsui, T., Tamura, S.: Adjacency on combinatorial polyhedra. Discrete Appl. Math. 56(2–3), 311–321 (1995)
McCarthy, D.J., Quintas, L.V.: The construction of minimal-line graphs with given automorphism group. In: Frank, F. (ed.) Topics in Graph Theory. Annals of the New York Academy of Science, vol. 328, pp. 144–152. New York Academy of Sciences, New York (1979)
Naddef, D., Pulleyblank, W.R.: Hamiltonicity and combinatorial polyhedra. J. Comb. Theory Ser. B 31(3), 297–312 (1981)
Robinson, D.J.S.: A Course in the Theory of Groups. Graduate Texts in Mathematics, 2nd edn. Springer, New York (1996)
Schulte, E., Williams, G.I.: Polytopes with preassigned automorphism groups. Discrete Comput. Geom. 54(2), 444–458 (2015)
Tran, T., Ziegler, G.M.: Extremal edge polytopes. Electron. J. Comb. 21(2), 2.57 (2014)
Acknowledgements
We would like to thank Selim Rexhep, a past Ph.D. student, for his many interesting questions and in particular the one asking for the existence of a convex polytope for any finite group. We thank also two anonymous reviewers for their useful comments on a preliminary version.
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Editor in Charge: Günter M. Ziegler, János Pach
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Doignon, JP. Any Finite Group is the Group of Some Binary, Convex Polytope. Discrete Comput Geom 59, 451–460 (2018). https://doi.org/10.1007/s00454-017-9945-0
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DOI: https://doi.org/10.1007/s00454-017-9945-0