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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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
- Nilpotent implies normalizer condition: Any nilpotent group satisfies the normalizer condition: it has no proper self-normalizing subgroup.
- Normalizer condition implies every maximal subgroup is normal
The proof follows by combining facts (1) and (2).