---
tag: V13JPPAZ4J
kind: concept
name: Holomorph (mathematics)
sources:
  - wikipedia: en:2841008@r1313036910
    locator: Holomorph (mathematics)
    note: adopted
    mention: wikipedia://en/2841008
    sections: [Summary]
same_as:
  wikidata: Q3139500
  wikipedia: Holomorph (mathematics)
inventory: [PYKM0XE5T7]
---

## 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|0CWXCWP0Y0]] $\operatorname{Aut}(G)$. 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|014E1DZQE8]] or as a permutation group.