# Induced class function from normal subgroup is zero outside the subgroup

## Statement

Suppose is a finite group and is a Normal subgroup (?) of . Suppose is a Class function (?) on with values in a field . Then, the Induced class function (?) takes the value zero at all elements not in .

## Related facts

- Induced class function from conjugacy-closed normal subgroup is index of subgroup times class function inside the subgroup and zero outside the subgroup: A conjugacy-closed normal subgroup is a normal subgroup where two elements of the subgroup conjugate in the whole group are also conjugate in the subgroup. In particular, any direct factor and any central factor is conjugacy-closed.