Normal subgroup contained in the hypercenter

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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				|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		|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