# Normal and self-centralizing implies normality-large

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., self-centralizing normal subgroup) must also satisfy the second subgroup property (i.e., normality-large normal subgroup)

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## Statement

Any Normal subgroup (?) of a group that is self-centralizing, is also normality-large: its intersection with every nontrivial normal subgroup is nontrivial.

## Proof

**Given**: A group , a normal subgroup , such that .

**To prove**: If is a nontrivial normal subgroup of such that is trivial, then is trivial.

**Proof**: Consider the commutator . This is contained both in and in , since they are both normal. Hence it is contained in , which is trivial. Thus, every element of commutes with every element of , so . Since is self-centralizing, we get . Since is trivial, we get that is trivial.