# Normalizer condition implies every maximal subgroup is normal

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., group satisfying normalizer condition) must also satisfy the second group property (i.e., group in which every maximal subgroup is normal)

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## Statement

Suppose is a group satisfying normalizer condition: it has no proper self-normalizing subgroup. In other words, every subgroup of satisfies the property that is *properly* contained in its normalizer .

Then, any maximal subgroup of is a normal subgroup.

## Proof

**Given**: A group satisfying the normalizer condition, a maximal subgroup of .

**To prove**: is normal in .

**Proof**: is a subgroup of containing . Since is maximal in , or . Since satisfies the normalizer condition, we cannot have . Thus, , and we obtain that is normal in .