# Covering group

From Groupprops

Template:Central extension property

## Contents

## Definition

### Symbol-free definition

A **covering group** of a group is defined as a central extension that is perfect as a group.

### Definition with symbols

Let be a group. A **covering group** for is a perfect group along with a map such that the kernel of lies inside the center of .

## Related group properties

### For a simple group

Any covering group of a simple group is a quasisimple group. Further, it is true that the universal covering group of a simple group is a finite quasisimple group.