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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and commutator-in-centralizer subgroup
View other subgroup property conjunctions | view all subgroup properties
- where denotes the commutator of two subgroups and denotes the center of .
- is a normal subgroup of and is trivial, i.e., is a commutator-in-centralizer subgroup.
A group has this property as a subgroup of itself if and only if it has nilpotency class two.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|Class two normal subgroup||normal subgroup of nilpotency class two|||FULL LIST, MORE INFO|
|Commutator-in-centralizer subgroup||commutator with whole group contained in centralizer|
|Hereditarily 2-subnormal subgroup||every subgroup is 2-subnormal in whole group||(via commutator-in-centralizer)||Commutator-in-centralizer subgroup|FULL LIST, MORE INFO|
Related group properties
A group has the property that for every group containing as a normal subgroup, is a commutator-in-center subgroup of , if and only if is a group whose inner automorphism group is central in automorphism group.