# Hypocenter

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

## Definition

### Symbol-free definition

The hypocenter of a group is defined as the limit of the lower central series of the group, that is, the intersection of all members of the lower central series.

## Properties

### Monotonicity

This subgroup-defining function is monotone, viz the image of any subgroup under this function is contained in the image of the whole group

The hypocenter of any subgroup is contained in the hypocenter of the whole group.

### 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 hypocenter of the hypocenter is the hypocenter. The image cum fixed-points of this are the hypocentral groups.

### 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

The hypocenter of the quotient of a group by its hypocenter is trivial.

## Properties satisfied

The hypocenter of any group is a fully characteristic subgroup.