2
$\begingroup$

I was in class today, and the teacher mentioned that the semidirect product given by $N \rtimes \textrm{Aut}(N)$ has a special name, but I couldn't understand what he said well enough to write it down. Does anyone know what it's called, what it's useful for, and where I could learn more about it?

$\endgroup$
5
$\begingroup$

This type of semidirect product is the so-called holomorph of the group $N$ and (at least the name) was introduced by Burnside. Apparently (after a quick google search), the following problem/question is the origin of the holomorph: "Is it possible to include any given group as a normal subgroup in some other group so that all the automorphisms of $N$ are restrictions of inner automorphisms of this large group?"

See here for more information and more uses.

$\endgroup$
1
  • $\begingroup$ Thank you, that's it! $\endgroup$ Oct 25 '19 at 21:39
1
$\begingroup$

It's called the outer semi-direct product.

$\endgroup$
3
  • $\begingroup$ I don't think the op is asking what the symbol means, but rather how this type of semidirect product is called as it has a special name. $\endgroup$
    – Con
    Oct 25 '19 at 21:28
  • $\begingroup$ The teacher may have been distinguishing between inner and outer semidirect products but, on balance, I think your interpretation is most likely to be correct. $\endgroup$
    – S. Dolan
    Oct 25 '19 at 21:33
  • $\begingroup$ Thanks for the help S. Dolan, but ThorWittich was correct - I know what semidirect products are and was looking for the specific name for the product. $\endgroup$ Oct 25 '19 at 21:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.