Epicenter

Definition
The epicenter (also called epicentre, precise center, or precise centre) of a group $$G$$, sometimes denoted $$Z^*(G)$$, is defined in the following equivalent ways:


 * 1) It is the intersection, over all defining ingredient::central extensions $$E$$ with quotient group $$G$$, of the images in the group of the center of the central extension group. Explicitly, it is the intersection $$\bigcap_{(E,\varphi)} \varphi(Z(E))$$ where $$(E,\varphi)$$ describes a central extension with quotient $$G$$, $$E$$ being the extension group and $$\varphi:E \to G$$ the quotient map.
 * 2) It is the unique largest normal subgroup of the group for which the quotient group is a defining ingredient::capable group.
 * 3) It is the subset of $$G$$ comprising those elements $$g \in G$$ such that, for all $$h \in G$$, $$g \wedge h$$ is the identity element of the exterior square of $$G$$.
 * 4) Suppose $$E$$ is a Schur covering group of $$G$$ with covering map $$\varphi:E \to G$$. The epicenter of $$G$$ is defined as $$\varphi(E)$$.

Properties not satisfied
The properties below are not always satisfied by the epicenter in a group. There may be groups where the epicenter satisfies the property.

Relation with other subgroup-defining functions

 * Epicentral series. Note that this is not the quotient-iterated series for the epicenter.

Larger subgroup-defining functions

 * contained in::Center

The computation problem
This command requires the Hap package, so if the package is not loaded, you need to load it as follows:

LoadPackage("hap");

To compute the epicenter of a group in GAP, the syntax is:

Epicenter(group);

where group could either be an on-the-spot description of the group or a name alluding to a previously defined group.

We can assign this as a value, to a new name, for instance:

zg := Epicenter(g);

where g is the original group and zg is the epicenter.