# Hypercenter

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Contents

## Definition

### Symbol-free definition

The **hypercenter** of a group is defined as the limit of the upper central series of the group.

### Definition with symbols

Let with varying over the cardinals, denote the upper central series of . Then, there exists a cardinal at which the upper central series stabilizes. The group for this is termed the **hypercenter**.

## Properties

### Reverse monotonicity

The **hypercenter** subgroup-defining function is reverse monotone. That is, if ≤ are groups, the hypercenter of contains the itnersection with of the hypercenter of .

### Idempotence

This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once

The hypercenter of the hypercenter is the hypercenter. The image cum fixed-point space for the hypercenter map is the property of being a hypercentral group.

### Quotient-idempotence

This subgroup-defining function is quotient-idempotent: taking the quotient of any group by the subgroup, gives a group where the subgroup-defining function yields the trivial subgroup

View a complete list of such subgroup-defining functions

If is a group and is the hypercenter of , then the hypercenter of is trivial.

## Subgroup properties satisfied

The hypercenter of any group is a strictly characteristic subgroup.