# Nilpotent implies every subgroup is subnormal

From Groupprops

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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## Contents

## 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

- Hypocentral implies every subgroup is descendant: In particular, since residually nilpotent implies hypocentral, every subgroup of a residually nilpotent group is descendant.

### Converse

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

- Every subgroup is subnormal not implies nilpotent
- Equivalence of definitions of finite nilpotent group