Holomorph (mathematics)

Summary

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group $G$, denoted $\operatorname{Hol}(G)$, is a group that simultaneously contains (copies of) $G$ and its automorphism group $\operatorname{Aut}(G)$. It provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. The holomorph can be described as a semidirect product or as a permutation group.