OFFSET
1,1
COMMENTS
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
REFERENCES
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.
LINKS
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
EXAMPLE
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.
PROG
(Magma) G<a, b>:=Group<a, b|a^2, b^3, (a*b)^7>; L:=LowIndexNormalSubgroups(G, 86016); for j in L do print (j`Index)/84+1; end for; // Bradley Brock, Nov 25 2012
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Richard Chapling (rc476(AT)cam.ac.uk), Aug 04 2010
EXTENSIONS
Corrected by Bradley Brock, Oct 01 2012
Entry revised by N. J. A. Sloane, Nov 25 2012
STATUS
approved