▁There ▁is ▁an ▁Ab elian ▁normal ▁subgroup ▁N ▁of ▁order ▁ 4 ▁( a ▁Klein ▁ 4 - sub group ) ▁which ▁is ▁not ▁contained ▁ ▁in ▁the ▁center ▁of ▁G . ▁Hence ▁ χ ▁is ▁induced ▁from ▁a ▁character ▁of ▁a ▁proper ▁subgroup ▁of ▁G ▁containing ▁N . ▁The ▁only ▁possibility ▁is ▁that ▁ χ ▁is ▁induced ▁from ▁a ▁linear ▁character ▁of ▁a ▁Sy low ▁ 2 - sub group ▁of ▁G . ▁ ▁Further ▁develop ments ▁Cl iff ord ' s ▁theorem ▁has ▁led ▁to ▁a ▁branch ▁of ▁representation ▁theory ▁in ▁its ▁own ▁right , ▁now ▁known ▁as ▁Cl iff ord ▁theory . ▁This ▁is ▁particularly ▁relevant ▁to ▁the ▁representation ▁theory ▁of ▁finite ▁sol v able ▁groups , ▁where ▁normal ▁sub groups ▁usually ▁ab ound . ▁For ▁more ▁general ▁finite ▁groups , ▁Cl iff ord ▁theory ▁often ▁allows ▁representation - the or etic ▁questions ▁to ▁be ▁reduced ▁to ▁questions ▁about ▁groups ▁that ▁are ▁close ▁( in ▁a ▁sense ▁which ▁can ▁be ▁made ▁precise ) ▁to ▁being ▁simple . ▁▁ ▁found ▁a ▁more ▁precise ▁version ▁of ▁this ▁result ▁for ▁the ▁restriction ▁of ▁ir reducible ▁unit ary ▁representations ▁of ▁locally ▁compact ▁groups ▁to ▁closed ▁normal ▁sub groups ▁in ▁what ▁has ▁become ▁known ▁as ▁the ▁" M ack ey ▁machine " ▁or ▁" M ack ey ▁normal ▁subgroup ▁analysis ". ▁ ▁References ▁ ▁Category : Re presentation ▁theory <0x0A> </s> ▁Mount ▁Tor bre ck ▁is ▁a ▁mountain ▁( e lev ation ▁ 1 5 1 6 ▁metres ) ▁in ▁Victoria , ▁Australia ▁ 1