Finite nilpotent implies every normal subgroup is part of a chief series

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Statement

Version for a group of prime power order

Version for a finite group

Suppose G is a Finite nilpotent group (?) and H is a normal subgroup of G. Then, there exists a Chief series (?) for G that has H as one of its members, i.e., there is a series:

\{ e \} = K_0 \le K_1 \le K_2 \le \dots \le K_r = G

such that each K_i is a normal subgroup of G and there exists some value of i for which K_i = H.