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I am reading about semidirect product in Dummit,Foots. I wonder the automorphism group of $\operatorname{Hol}(\mathbb{Z}_n)$, that is, $\mathbb{Z}_n\rtimes\mathbb{Z}_n^×$ with an operation $(a,b)*(c,d)=(a+bc,bd)$.

Generally, I hope to know about Holomorph of Holomorph, or the relation between $\operatorname{Hol}(G)$, $\operatorname{Hol}(K)$ and $\operatorname{Hol}(G×K)$. Is there a good theorem or any reference?

Samuel Adrian Antz
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LWW
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    $a+bc$ here is not well-defined – Maxime Ramzi Mar 25 '18 at 09:11
  • Oh, there are typos. Edited! – LWW Mar 25 '18 at 09:17
  • And so more generally you define $Hol(G)$ as the semi-direct product of $G$ by $Aut(G)$ ? – Maxime Ramzi Mar 25 '18 at 09:52
  • The automorphism group of a direct prodcut can be quite complicated. See ere for example. As for ${\rm Hol}({\mathbb Z}/n{\mathbb Z})$, I think it is complete when $n$ is odd, but has a small outer automorphism group when $n$ is even. – Derek Holt Mar 25 '18 at 11:36
  • Max / Yes. Derek Holt / Thank you. Can you let me know why are you think so? – LWW Mar 25 '18 at 12:26
  • @LWW, the proof of the completeness of $Hol(C_n)$ for odd $n$ is in the answer to this question: https://math.stackexchange.com/questions/2863363/is-the-statement-that-operatornameaut-operatornameholz-n-cong-oper – Chain Markov Feb 22 '19 at 09:21

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