Nilpotent 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 must also satisfy the second group property
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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Maximal subgroup (?)) must also satisfy the second subgroup property (i.e., Normal subgroup (?)). In other words, every maximal subgroup of nilpotent group is a normal subgroup of nilpotent group.
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Contents
Statement
Propertytheoretic statement
As am implication of group properties: The property of being a Nilpotent group (?) is stronger than the property of being a Group in which every maximal subgroup is normal (?).
As an implication of subgroup properties: In a nilpotent group, the property of being a maximal subgroup is stronger than the property of being a normal subgroup.
Verbal statement
In a nilpotent group, every Maximal subgroup (?) is normal.
Related facts
More on nilpotent groups
Further information: maximal subgroup in nilpotent group, maximal subgroup in finite nilpotent group, maximal subgroup of group of prime power order, equivalence of definitions of maximal subgroup of group of prime power order
It's further true that since the maximal subgroup is normal, it is maximal normal, so the quotient is simple. Since a quotient of a nilpotent group is nilpotent, and the only simple nilpotent groups are abelian, we conclude that every maximal subgroup is normal with abelian quotient (specifically, the quotient is a cyclic group of prime order).
Subgroups of small index in other types of groups
 Maximal subgroup has prime power index in finite solvable group
 Subgroup of index equal to least prime divisor of group order is normal
Subgroups of small order
 Minimal normal implies central in nilpotent
 Normal of order equal to least prime divisor of group order implies central
Facts used
 Nilpotent implies normalizer condition: Any nilpotent group satisfies the normalizer condition: it has no proper selfnormalizing subgroup.
 Normalizer condition implies every maximal subgroup is normal
Proof
The proof follows by combining facts (1) and (2).