A343822 -id:A343822 - cp's OEIS frontend

A179982 Values of the genus g for which there exists a compact Riemann surface of genus g admitting an automorphism group of order 84(g-1), the maximum possible, also known as the Hurwitz bound.

3, 7, 14, 17, 118, 129, 146, 385, 411, 474, 687, 769, 1009, 1025, 1459, 1537, 2091, 2131, 2185, 2663, 3404, 4369, 4375, 5433, 5489, 6553, 7201, 8065, 8193, 8589, 11626, 11665

Offset: 1

Richard Chapling (rc476(AT)cam.ac.uk), Aug 04 2010

It is known that the sequence is infinite, and also that the sequence of g for which the Hurwitz bound is not attained is infinite. No generating formula is known.
The group in question is a Hurwitz group, and so is generated by x,y with x^2 = y^3 = (xy)^7 = 1.
For k in A343821 (not in A343822), k!/168 + 1 is in this sequence since the alternating group A_k is a Hurwitz group. In particular, k!/168 + 1 is a term for all k >= 168. - Jianing Song, Jul 13 2021

			g=3 is satisfied by the Klein quartic x^3 * y + y^3 * z + z^3 * x = 0; the group is isomorphic to PSL(2,Z_7), the projective special linear group of 2 X 2 matrices with entries modulo 7.

Marston Conder, Hurwitz groups: a brief survey. Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 359-370.
Gareth A. Jones and David Singerman, Complex Functions: An algebraic and geometric viewpoint, Cambridge University Press, 1987, pp. 263-266.
E. B. Vinber and O. V. Shvartsman, Riemann surfaces, Journal of Mathematical Sciences, 14, #1 (1980), 985-1020. Riemann surfaces, Algebra, Topologiya, Geometriya, Vol. 16 (Russian), pp. 191-245, 247 (errata insert), VINITI , Moscow, 1978.

Jeffrey M. Cohen, On Hurwitz extensions by PSL_2(7), Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 3, 395-400.
Marston Conder, The genus of compact Riemann surfaces with maximal automorphism group, J. Algebra 108 (1987), no. 1, 204-247.
Chih-han Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969) 13-42.
Wikipedia, Hurwitz Group

Cf. A343821, A343822, A346293.

G:=Group; L:=LowIndexNormalSubgroups(G,86016); for j in L do print (j`Index)/84+1; end for; // Bradley Brock, Nov 25 2012

Corrected by Bradley Brock, Oct 01 2012

Entry revised by N. J. A. Sloane, Nov 25 2012

A343821 Numbers k such that the alternating group A_k is a Hurwitz group.

Original entry on oeis.org

15, 21, 22, 28, 29, 35, 36, 37, 42, 43, 45, 49, 50, 51, 52, 56, 57, 58, 63, 64, 65, 66, 70, 71, 72, 73, 77, 78, 79, 80, 81, 84, 85, 86, 87, 88, 91, 92, 93, 94, 96, 98, 99, 100, 101, 102, 105, 106, 107, 108, 109, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122

Offset: 1

Eric M. Schmidt, Apr 30 2021

nonn

This sequence contains every k > 167 [Conder].

This sequence is found in Section 2 of the Gordejuela and Martínez paper, which has a slight error: 86 occurs twice and 87 is missing.

Eric M. Schmidt, Table of n, a(n) for n = 1..150
Marston D. E. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), pp. 75-86.
J. J. Etayo Gordejuela and E. Martínez, Alternating groups, Hurwitz groups and H^*-groups, Journal of Algebra, 283 (2005), pp. 327-349.

Complement of A343822.

Cf. A179982.

cp's OEIS Frontend

A179982 Values of the genus g for which there exists a compact Riemann surface of genus g admitting an automorphism group of order 84(g-1), the maximum possible, also known as the Hurwitz bound.

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