On Metabelian Groups of Automorphisms of Compact Riemann Surfaces

Supersoluble groups of automorphisms of compact Riemann surfaces

“…From above, given n 5= 2, there exists a supersoluble group G of automorphisms of order 18(g -1) of a compact Riemann surface of genus g with g = 3" + l. Thus G has generators x x ,x 2 satisfying x\ = x\ = (xxX 2 ) ls = 1. By the Lemma 3.3 G/G' = Z 6 and so there is a homomorphism T: G -* ((p) c Aut(Z/), with the images of the generators x u x 2 of order 2 and 3 respectively being 0 3 and (f> 2 .…”

“…For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, /7-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, 5F) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e.…”

Metabelian Groups Acting on Compact Riemann Surfaces

“…A firaite group O is said tobe a (k,l,m)-groupif it cara be generated by two elements of order le arad 1 witose product Itas order m. From [2] it foliows titat tite problem of describirag metabeliara groups titat can occur as gronps of automorphisms of order 16(g -1) of a compact Riemaran sUrface of genus g = 2 Ls eqnivalent to tite pure]y group titeoretical problem of flndirag all finite metabeliara (2,4,8) Proof. O Ls generated by elements a arad b of order 2 arad 4 respectively whose product itas order 8.…”

“…Partial results have been obtairaed by Citetiya and Patra in [2] witere titey proved that a metabeliara group of automorphisms of a compact Riemann surface of geraus y = 2 (y # 2,3,5) itas at most 16(9 - 1) elements. Ira titis paper we go much furtiter by describing exactly titose values of y for witicit titis honrad is sitarp, flndirag tite presentatioras of alí corresponding groups by means of deflrairag generators arad relatioras arad flraaliy citaracterizirag ira terms of titese groups titose surfaces whicit are symmetric i.e., admitting ara araticonformal involutiora.…”

“…sequences of g for whicit tite correspondirag bourads are sItarp were fourad ira [10,3,4,5,6,11,12,13,15,16] amorag tIte vaste literature, [18,19], [7], [1,8] and [2] respectively. l 1lowever tite problem of flradirag for givera Y all g and all groups for witicit tite correspondirag honrad is acitieved is ratiter difficult.…”