Nilpotent implies intersection of normal subgroup with upper central series is strictly ascending till the subgroup is reached

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Suppose G is a nilpotent group and H is a normal subgroup of G. Suppose Z^{(i)}(G) denotes the i^{th} member of the upper central series of G, i.e., Z^{(0)}(G) is the trivial subgroup, Z^{(1)}(G) is the center, and Z^{(i+1)}(G)/Z^{(i)}(G) is the center of G/Z^{(i)}(G).

Let r be the smallest nonnegative integer such that H \le Z^{(r)}(G). Then H \cap Z^{(i)}(G) is a proper subgroup of H \cap Z^{(i+1)}(G) for all 0 \le i < r.

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