Normal subgroup of prime index
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup of prime index
View other subgroup property conjunctions | view all subgroup properties
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and maximal subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A normal maximal subgroup or normal subgroup of prime index is defined in the following equivalent ways:
- It is a normal subgroup that is also maximal among proper subgroups: it is not contained in any strictly bigger proper subgroup.
- It is a normal subgroup and its index is a prime number.
Relation with other properties
Stronger properties
- Subgroup of index two: For full proof, refer: Subgroup of index two is normal
- Subgroup of least prime index: For full proof, refer: Subgroup of least prime index is normal
Weaker properties
- Maximal normal subgroup: Note that the properties are equivalent for a solvable group, and more generally, for an imperfect group.
- Subgroup of prime index: Note that the properties are equivalent for a nilpotent group. For full proof, refer: Nilpotent implies every maximal subgroup is normal
- Maximal subgroup: Note that the properties are equivalent for a nilpotent group. For full proof, refer: Nilpotent implies every maximal subgroup is normal