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 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 .
- is abelian iff .
- Consider the radical of
\begin{align*}
R = { a \in A : c_{0}(a,A) = 0 }
\end{align*}
\begin{align*}
c_{0}(a,b) = \varepsilon_{0}(a,b) - \varepsilon_{0}(b,a)
\end{align*}