Holomorph (mathematics)

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group ${\displaystyle G}$, denoted ${\displaystyle \operatorname {Hol} (G)}$, is a group that simultaneously contains (copies of) ${\displaystyle G}$ and its automorphism group ${\displaystyle \operatorname {Aut} (G)}$. 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.