Centralizer of divisibility-closed subgroup is completely divisibility-closed in nilpotent group

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Statement

Suppose G is a nilpotent group and H is a divisibility-closed subgroup of G. Then, the centralizer C_G(H) of H in G is a completely divisibility-closed subgroup of G.

In particular, this shows that the property of being a divisibility-closed subgroup of nilpotent group is a centralizer-closed subgroup property, and also that the property of being a completely divisibility-closed subgroup of nilpotent group is a centralizer-closed subgroup property.

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