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
Contents
Definition
The epicenter (also called epicentre, precise center, or precise centre) of a group , sometimes denoted
, is defined in the following equivalent ways:
- It is the intersection, over all central extensions
with quotient group
, of the images in the group of the center of the central extension group. Explicitly, it is the intersection
where
describes a central extension with quotient
,
being the extension group and
the quotient map.
- It is the unique largest normal subgroup of the group for which the quotient group is a capable group.
- It is the subset of
comprising those elements
such that, for all
,
is the identity element of the exterior square of
.
- Suppose
is a Schur covering group of
with covering map
. The epicenter of
is defined as
.
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 ![]() |
---|---|---|
trivial subgroup | capable group | ![]() ![]() ![]() |
whole group | epabelian group | If ![]() ![]() ![]() ![]() ![]() |
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
Computation
The computation problem
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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.