Editing Transfer (group theory) (section)

==Commutator subgroup==

If ''G'' is finitely generated, the [[commutator subgroup]] ''G''′ of ''G'' has finite index in ''G'' and ''H=G''′, then the corresponding transfer map is trivial. In other words, the map sends ''G'' to 0 in the abelianization of ''G''′. This is important in proving the [[principal ideal theorem]] in [[class field theory]].<ref name=S122>Serre (1979) p.122</ref>  See the [[Emil Artin]]-[[John Tate (mathematician)|John Tate]] ''Class Field Theory'' notes.