▁integer ▁e ▁div ides ▁the ▁index ▁ ▁though ▁the ▁proof ▁of ▁this ▁fact ▁requires ▁some ▁use ▁of ▁Sch ur ' s ▁theory ▁of ▁project ive ▁representations . ▁ ▁Proof ▁of ▁Cl iff ord ' s ▁theorem ▁ ▁The ▁proof ▁of ▁Cl iff ord ' s ▁theorem ▁is ▁best ▁explained ▁in ▁terms ▁of ▁modules ▁( and ▁the ▁module - the or etic ▁version ▁works ▁for ▁ir reducible ▁mod ular ▁representations ). ▁Let ▁F ▁be ▁a ▁field , ▁V ▁be ▁an ▁ir reducible ▁F [ G ] - module , ▁V N ▁be ▁its ▁restriction ▁to ▁N ▁and ▁U ▁be ▁an ▁ir reducible ▁F [ N ] - sub module ▁of ▁V N . ▁For ▁each ▁g ▁in ▁G , ▁U . g ▁is ▁an ▁ir reducible ▁F [ N ] - sub module ▁of ▁V N , ▁and ▁▁ ▁is ▁an ▁F [ G ] - sub module ▁of ▁V , ▁so ▁must ▁be ▁all ▁of ▁V ▁by ▁ir redu ci b ility . ▁Now ▁V N ▁is ▁expressed ▁as ▁a ▁sum ▁of ▁ir reducible ▁sub modules , ▁and ▁this ▁expression ▁may ▁be ▁ref ined ▁to ▁a ▁direct ▁sum . ▁The ▁proof ▁of ▁the ▁character - the or etic ▁statement ▁of ▁the ▁theorem ▁may ▁now ▁be ▁completed ▁in ▁the ▁case ▁F ▁= ▁C . ▁Let ▁ χ ▁be ▁the ▁character ▁of ▁G ▁ ▁afford ed ▁by ▁V ▁and ▁ μ ▁be ▁the ▁character ▁of ▁N ▁afford ed ▁by ▁U . ▁For ▁each ▁g ▁in ▁G , ▁the ▁C [ N ] - sub module