Nilpotent implies every subgroup is subnormal

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., nilpotent group) must also satisfy the second group property (i.e., group in which every subgroup is subnormal)
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Statement

Verbal statement

In a nilpotent group, every subgroup is subnormal. Moreover, the Subnormal depth (?) of any subgroup is bounded by the Nilpotence class (?) of the group.

Related facts

Similar facts

Converse

The converse is not true in general, but is true for finite groups:

Other weaker conditions true for nilpotent groups