Hypercenter

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 $$Z^{\alpha}(G)$$ with $$\alpha$$ varying over the cardinals, denote the upper central series of $$G$$. Then, there exists a cardinal $$\alpha$$ at which the upper central series stabilizes. The group $$Z^{\alpha}(G)$$ for this $$\alpha$$ is termed the hypercenter.

Reverse monotonicity
The hypercenter subgroup-defining function is reverse monotone. That is, if $$H$$ &le; $$G$$ are groups, the hypercenter of $$H$$ contains the itnersection with $$H$$ of the hypercenter of $$G$$.

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.

If $$G$$ is a group and $$S$$ is the hypercenter of $$G$$, then the hypercenter of $$G/S$$ is trivial.

Subgroup properties satisfied
The hypercenter of any group is a strictly characteristic subgroup.