Central extension

Definition

A central extension is a group extension where the base normal subgroup is a central subgroup, i.e., it is contained in the center of the whole group.

Central extensions are significant because the central extensions for a given normal subgroup (in this case central subgroup) and quotient group are classified by the second cohomology group for trivial group action of the quotient group on the normal subgroup.

Properties

• Stem extension
• Split central extension
• Perfect central extension
• Universal central extension