[[Group theory MOC]]
# Holomorph group

Let $G$ be a group.
The **holomorph** $\opn{Hol}G$ of $G$ is the [[semidirect product]] with its [[Automorphism|automorphism group]] #m/def/group
$$
\begin{align*}
\opn{Hol} G = G \rtimes  \Aut G
\end{align*}
$$
## Properties

- $\opn{Hol} G$ acts faithfully on $G$ by combining the [[regular group action]] with the canonical action of $\Aut G$.


#
---
#state/develop | #lang/en | #SemBr