# Normal subgroup whose derived subgroup equals its intersection with whole derived subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup whose commutator subgroup equals its intersection with whole commutator subgroup

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## Definition

A subgroup of a group is termed a **normal subgroup whose commutator subgroup equals its intersection with whole commutator subgroup** if it satisfies the following equivalent conditions:

- is a normal subgroup of and .
- .