Hypocenter

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

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

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.