Normal subgroup contained in the hypercenter

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup of a group is termed a normal subgroup contained in the hypercenter if it is a normal subgroup and it is contained in the hypercenter of the whole group (the hypercenter is the subgroup at which the transfinite upper central series eventually terminates).

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup of nilpotent group |FULL LIST, MORE INFO
central subgroup |FULL LIST, MORE INFO
subgroup of abelian group Normal subgroup of nilpotent group|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
amalgam-characteristic subgroup normal subgroup contained in the hypercenter is amalgam-characteristic Amalgam-strictly characteristic subgroup|FULL LIST, MORE INFO
normal subgroup satisfying the subgroup-to-quotient powering-invariance implication if the whole group and the normal subgroup are powered over a prime, so is the quotient group. normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication any finite non-nilpotent group as a subgroup of itself |FULL LIST, MORE INFO