Holomorph (mathematics)

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