Epicenter

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

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 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 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 $\varphi(E)$.

Equivalence of definitions

Further information: equivalence of definitions of epicenter

Particular cases

Group property	Computing the epicenter for such a group
finite group	the epicenter can also be defined as the intersection of the kernels of all projective representations over the complex numbers, where the kernel of a projective representation is understood to mean the kernel of the corresponding homomorphism to the projective general linear group.
finite abelian group	characterization of epicenter of finite abelian group -- the epicenter can be described and computed explicitly.

Related group properties

Epicenter	Corresponding group property	What it means for a group $G$
trivial subgroup	capable group	$G$ is isomorphic to the inner automorphism group of some group, i.e., $G \cong K/Z(K)$ for some group $K$.
whole group	epabelian group	If $G$ is isomorphic to the quotient of some group $K$ by a central subgroup $H$ of $K$, then $K$ is abelian.
center	unicentral group	It means that the quotient by any proper subgroup of the center is not capable.

Subgroup properties

Properties satisfied

Property	Meaning	Proof of satisfaction
normal subgroup	invariant under all inner automorphisms	follows from being characteristic, also follows from being central
characteristic subgroup	invariant under all automorphisms	follows from subgroup-defining function value is characteristic
central subgroup	contained inside the center

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.

Property	Meaning	Proof of satisfaction
fully invariant subgroup	invariant under all endomorphisms	epicenter not is fully invariant

Relation with other subgroup-defining functions

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

Larger subgroup-defining functions

• Center

Computation

The computation problem

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GAP command

The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:Epicenter
View other GAP-computable subgroup-defining functions

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.