( G ) ▁is ▁defined ▁to ▁be ▁the ▁subgroup ▁generated ▁by ▁all ▁the ▁Sy low ▁sub groups ▁of ▁G ▁for ▁the ▁prime ▁number ▁p . ▁This ▁is ▁a ▁normal ▁subgroup , ▁and ▁the ▁parameter ▁n ▁is ▁defined ▁as ▁the ▁minimum ▁number ▁of ▁gener ators ▁of ▁ ▁G / p ( G ). ▁ ▁Then ▁for ▁the ▁case ▁of ▁C ▁the ▁project ive ▁line ▁over ▁K , ▁the ▁conject ure ▁states ▁that ▁G ▁can ▁be ▁real ised ▁as ▁a ▁Gal ois ▁group ▁of ▁L , ▁un ram ified ▁outside ▁S ▁containing ▁s ▁+ ▁ 1 ▁points , ▁ ▁if ▁and ▁only ▁if ▁ ▁n ▁ ≤ ▁s . ▁ ▁This ▁was ▁proved ▁by ▁Ray na ud . ▁ ▁For ▁the ▁general ▁case , ▁proved ▁by ▁Har b ater , ▁let ▁g ▁be ▁the ▁genus ▁of ▁C . ▁Then ▁G ▁can ▁be ▁real ised ▁if ▁and ▁only ▁if ▁ ▁n ▁ ≤ ▁s ▁+ ▁ 2 ▁g . ▁ ▁References ▁ ▁External ▁links ▁▁▁▁ ▁A ▁lay man ' s ▁perspective ▁of ▁Ab hy ank ar ' s ▁conject ure ▁from ▁Pur due ▁University ▁ ▁Category : Al gebra ic ▁curves ▁Category : G alo is ▁theory ▁Category : The or ems ▁in ▁abstract ▁algebra ▁Category : Con ject ures ▁that ▁have ▁been ▁proved <0x0A> </s> ▁Thomas ▁Kelly ▁( Oct ober ▁ 1 8 3 3 ▁– ▁ 1 8 9 3 ) ▁was ▁a ▁lawyer , ▁judge ▁and ▁political ▁figure ▁in ▁Prince ▁Edward ▁Island . ▁He ▁represented ▁ 5 th ▁Prince ▁in ▁the ▁Legisl ative