Central Extension

An extension

is a central extension if $\iota(A)$ is contained in the center $Z(E)$. Equivalently, the kernel $A$ commutes with every element of $E$.

Central extensions are a special case of group extensions in which the “added part” sits centrally. In particular, $A$ is necessarily abelian , since every subgroup of the center is abelian.

Examples:

• The quaternion group fits into a central extension $1\to \{\pm 1\}\to Q_8\to Q_8/\{\pm1\}\to 1$.
• If $E$ is abelian, then every extension $1\to A\to E\to Q\to 1$ is central.
• For any group $G$, the quotient map $G\to G/Z(G)$ exhibits $G$ as a central extension of $G/Z(G)$ by $Z(G)$.