Central group extension

Central extension of an abelian group

Let be an abelian group and consider a central extension

Then is nilpotent with its commutator subgroup central #m/thm/group since

and given a -section of we have the associated commutator map

which is alternating -bilinear and independent of .¹

Properties

In what follows, if is a subgroup let .

1. is abelian iff .
2. Consider the radical of

Then . 3. The associated 2-cycle and associated commutator are related by

that is, is the antisymmetrization of .

Special cases

• Cyclic central extension of a free abelian group
  • 2 central extension of a free abelian group
• 2 central extension of an elementary abelian 2-group (includes extraspecial 2-groups)

#state/tidy | #lang/en | #SemBr