▁the ▁equivalence ▁classes ▁of ▁ir reducible ▁representations ▁of ▁N . ▁In ▁particular ▁the ▁number ▁of ▁pair wise ▁non is omorphic ▁summ ands ▁is ▁no ▁greater ▁than ▁the ▁index ▁of ▁N ▁in ▁G . ▁ ▁Cl iff ord ' s ▁theorem ▁yields ▁information ▁about ▁the ▁restriction ▁of ▁a ▁complex ▁ir reducible ▁character ▁of ▁a ▁finite ▁group ▁G ▁to ▁a ▁normal ▁subgroup ▁N . ▁If ▁ μ ▁is ▁a ▁complex ▁character ▁of ▁N , ▁then ▁for ▁a ▁fixed ▁element ▁g ▁of ▁G , ▁another ▁character , ▁ μ ( g ), ▁of ▁N ▁may ▁be ▁constructed ▁by ▁setting ▁ ▁for ▁all ▁n ▁in ▁N . ▁The ▁character ▁ μ ( g ) ▁is ▁ir reducible ▁if ▁and ▁only ▁if ▁ μ ▁is . ▁Cl iff ord ' s ▁theorem ▁states ▁that ▁if ▁ χ ▁is ▁a ▁complex ▁ir reducible ▁character ▁of ▁G , ▁and ▁ μ ▁is ▁an ▁ir reducible ▁character ▁of ▁N ▁with ▁▁ ▁then ▁ ▁where ▁e ▁and ▁t ▁are ▁positive ▁integers , ▁and ▁each ▁gi ▁is ▁an ▁element ▁of ▁G . ▁The ▁integers ▁e ▁and ▁t ▁both ▁divide ▁the ▁index ▁[ G : N ]. ▁The ▁integer ▁t ▁is ▁the ▁index ▁of ▁a ▁subgroup ▁of ▁G , ▁containing ▁N , ▁known ▁as ▁the ▁in ert ial ▁subgroup ▁of ▁ μ . ▁This ▁is ▁ ▁and ▁is ▁often ▁denoted ▁by ▁ ▁The ▁elements ▁gi ▁may ▁be ▁taken ▁to ▁be ▁represent atives ▁of ▁all ▁the ▁right ▁cos ets ▁of ▁the ▁subgroup ▁I G ( μ ) ▁in ▁G . ▁ ▁In ▁fact , ▁the