Abstract
For \(q\ge 4\), we show that an almost simple group G with socle \(PSL\left( 2,q\right) \) is the automorphism group of at least one abstract chiral polyhedron if and only if G is not one of \(PSL\left( 2,q\right) \), \(PGL\left( 2,q\right) \), \(M\left( 1,9\right) \) or \(P\Sigma L\left( 2,4\right) \cong PGL\left( 2,5\right) \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For instance, if \(q=81=3^4\), there are two groups \(M\left( 1,81\right) \) and \(M\left( 2,81\right) \), one for each divisor \(d\not =4\) of 4; there are two extensions of \(PSL\left( 2,81\right) \) by a field automorphism: \(P\Sigma L\left( 2,81\right) =PSL\left( 2,81\right) \rtimes \left\langle \varphi \right\rangle \) and \(PSL\left( 2,81\right) \rtimes \left\langle \varphi ^2\right\rangle =PSL\left( 2,81\right) \rtimes \left\langle \beta \right\rangle \), one for each divisor \(d\not =4\) of 4; finally, there are two extensions of \(PGL\left( 2,81\right) \) by a field automorphism: \(P\Gamma L\left( 2,81\right) =PGL\left( 2,81\right) \rtimes \left\langle \varphi \right\rangle \) and \(PGL\left( 2,81\right) \rtimes \left\langle \varphi ^2\right\rangle =PGL\left( 2,81\right) \rtimes \left\langle \beta \right\rangle \), one for each divisor \(d\not =4\) of 4. So, together with \(PSL\left( 2,81\right) \) and \(PGL\left( 2,81\right) \), we have listed all the eight subgroups \(PSL\left( 2,81\right) \le G\le P\Gamma L\left( 2,81\right) \) corresponding to the subgroups of \(C_4\times C_2\).
References
Brooksbank, P.A., Leemans, D.: Polytopes of large rank for \(\text{ PSL }(4,{\mathbb{F}_q})\). J. Algebra 45, 390–400 (2016)
Brooksbank, P.A., Vicinsky, D.A.: Three-dimensional classical groups acting on polytopes. Discrete Comput. Geom. 44(3), 654–659 (2010)
Conder, M., Hubard, I., O’Reilly-Regueiro, E., Pellicer, D.: Construction of chiral 4-polytopes with alternating or symmetric automorphism group. J. Algebr. Comb. 42(1), 225–244 (2015)
Conder, M., Hubard, I., O’Reilly-Regueiro, E., Pellicer, D.: Construction of chiral polytopes of large rank with alternating or symmetric automorphism group. (In preparation)
Conder, M., Potočnik, P., Širáň, J.: Regular hypermaps over projective linear groups. J. Aust. Math. Soc. 85(2), 155–175 (2008)
Connor, T., De Saedeleer, J., Leemans, D.: Almost simple groups with socle \(\text{ PSL }(2, q)\) acting on abstract regular polytopes. J. Algebra 423, 550–558 (2015)
Connor, T., Leemans, D.: Algorithmic enumeration of regular maps. ARS Math. Contemp. 10, 211–222 (2016)
Connor, T., Leemans, D., Mixer, M.: Abstract regular polytopes for the O’Nan group. Int. J. Algebra Comput. 24, 59–68 (2014)
Dickson, L.E.: Linear Groups: With An Exposition of the Galois Field Theory. With an Introduction by W. Magnus. Dover Publications Inc., New York (1958)
Elliott, D., Leemans, D., O’Reilly-Regueiro, E.: Chiral polyhedra and projective lines. Int. J. Math. Stat. 16(1), 45–52 (2015)
Fernandes, M.-E., Leemans, D.: Polytopes of high rank for the symmetric groups. Adv. Math. 228, 3207–3222 (2011)
Fernandes, M.-E., Leemans, D., Mixer, M.: All alternating groups \(A_n\) with \(n\ge 12\) have polytopes of rank \(\lfloor \frac{n-1}{2}\rfloor \). SIAM J. Discrete Math. 26(2), 482–498 (2012)
Fernandes, M.-E., Leemans, D., Mixer, M.: Polytopes of high rank for the alternating groups. J. Comb. Theory Ser. A 119, 42–56 (2012)
Galbraith, S.D.: Mathematics of Public Key Cryptography. Cambridge University Press, Cambridge (2012)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1980)
Hartley, M.I., Hubard, I., Leemans, D.: Two atlases of abstract chiral polytopes for small groups. ARS Math. Contemp. 5(2), 371–382 (2012)
Hartley, M.I., Hulpke, A.: Polytopes derived from sporadic simple groups. Contrib. Discrete Math. 5(2), 106–118 (2010)
Hubard, I., Leemans, D.: Chiral polytopes and Suzuki simple groups. In: Connelly, R. (ed.) Rigidity and Symmetry, Fields Institute Communications 70:155–175 (2014)
Kiefer, A., Leemans, D.: On the number of abstract regular polytopes whose automorphism group is a Suzuki simple group \({S}z(q)\). J. Combin. Theory Ser. A 117(8), 1248–1257 (2010)
Leemans, D.: The residually weakly primitive geometries of the dihedral groups. ATTI Semin. Mat. Fis. Univ. Modena 48(1), 179–190 (2000)
Leemans, D.: Almost simple groups of Suzuki type acting on polytopes. Proc. Amer. Math. Soc. 134, 3649–3651 (2006)
Leemans, D., Mixer, M.: Algorithms for classifying regular polytopes with a fixed automorphism group. Contrib. Discrete Math. 7(2), 105–118 (2012)
Leemans, D., Schulte, E.: Groups of type \({L}_{2}(q)\) acting on polytopes. Adv. Geom. 7, 529–539 (2007)
Leemans, D., Schulte, E.: Polytopes with groups of type \({PGL}(2, q)\). ARS Math. Contemp. 2, 163–171 (2009)
Leemans, D., Schulte, E., Van Maldeghem, H.: Groups of Ree type in characteristic 3 acting on polytopes. Preprint (2015). arXiv:1501.01125
Leemans, D., Vauthier, L.: An atlas of abstract regular polytopes for small groups. Aequationes Math. 72, 313–320 (2006)
McMullen, P., Schulte, E.: Abstract Regular Polytopes, Volume 92 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2002)
Schulte, E., Ivić Weiss, A.: Chiral polytopes. In: Applied Geometry and Discrete Mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 493–516. American Mathematical Society, Providence (1991)
Acknowledgments
We thank Steven Galbraith who pointed us to the reference of Lemma 4.7. This work has been financially supported by Marsden Grant UOA1218 from the Royal Society of New Zealand.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leemans, D., Moerenhout, J. Chiral polyhedra arising from almost simple groups with socle \(PSL\left( 2,q\right) \) . J Algebr Comb 44, 293–323 (2016). https://doi.org/10.1007/s10801-016-0669-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-016-0669-7