= Central translations in a linear semidirect product
{title2=$(e,v)\in Z(V\rtimes H)\Longleftrightarrow\rho(h)v=v\ \text{for all }h$}
Let an <abelian group> $V$ carry an action $\rho$ of a <group> $H$ by automorphisms, and use multiplication $(h,v)(k,w)=(hk,v+\rho(h)w)$. Comparing $(e,v)(h,w)$ and $(h,w)(e,v)$ shows that a translation is central precisely when its vector is fixed by the entire action. If $H$ is itself abelian, the whole semidirect product is abelian exactly when the action is trivial. For $H=(\mathbb R,+)$, $V=\mathbb R^2$, and determinant-one linear actions fixing the first coordinate vector, $\rho(x)=\begin{pmatrix}1&r(x)\\0&1\end{pmatrix}$, where $r$ is an <additive function>. Without a regularity assumption, it need not be $r(x)=cx$.
Back to article page
