# Why is this group called “The Holomorph of a group”

Many years ago I found in google the notation "Holomorph of group". It is the semi direct product of $G$ with $Aut(G)$. Why is the term "Holomorph" used here, while it is usually used for complex analytic functions? More information on this object is very appreciated.

• I don't know anything about holomorph groups, but it's probably a word of Greek origin, meaning something like "complete form", "completely formed" or so.. – Qfwfq Jan 17 '15 at 8:48
• One nice way to consider the holomorph is as the normalizer of the left regular representation in the group of permutations of elements. – S. Carnahan Jan 17 '15 at 10:18
• according to Wikipedia, the name holomorph was introduced by Briot and Bouquet --- en.citizendium.org/wiki/Holomorphic_function – Carlo Beenakker Jan 17 '15 at 16:03

I'm not a history expert, but according to Miller, Blichfeldt and Dickson: "Theory and applications of finite groups" (1916), footnote p. 46: "The concept of holomorph was used by many early writers, but the term was introduced by W. Burnside in the first edition of his Theory of Groups, 1897, p. 228."

• Yes, what was introduced by the people Carlos Beenakker mentions was the term "holomorphic function" ( at least according to Wikipedia). – Geoff Robinson Jan 17 '15 at 18:38