# Normal subgroup contained in the hypercenter

From Groupprops

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]

## Contents

## 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 |