Normalizer condition implies every maximal subgroup is normal
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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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 .