If is a finite group and is a normal subgroup of . Prove that there is a composition series of , one of whose terms is .

Recall that every finite group has a composition series.

Definition: A composition series for is a series of subgroups of : is a normal subgroup of is a normal subgroup of is a normal subgroup of is a normal subgroup of such that is a normal subgroup of and is simple.

Proof:

is a finite group, so has a composition series. And is a normal subgroup of , by the definition of a composition series, must be a term of the series.

Does this complete the proof? If not, please help. Thank you.