# 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 | is isomorphic to the inner automorphism group of some group, i.e., for some group . |

whole group | epabelian group | If is isomorphic to the quotient of some group by a central subgroup of , then 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

## 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) isEpicenter

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.