# Inner wreath product

the standard definition of inner semidirect product says that $$G$$ is a semidirect product of $$K$$ and $$H$$ if $$G = KH$$, for some subgroup $$K$$ and $$H$$ of $$G$$ such that $$K$$ is normal and $$K\cap H =\{H\}$$.

Usually wreath product of groups $$K\wr H$$ is defined as and external semidirect product with $$K^n\rtimes_\Phi H$$, where $$\Phi$$ is a homomorphism $$H\to Aut(K^n)$$.

Is it possible to define an inner wreath product without talking about the homomorphism, i.e., just by decomposing a group into some subgroups satisfying some conditions?

• You've already done it: the wreath product is a semi-direct product, and semi-direct products can be discussed 'internally'. (Also, shouldn't $H$ be equipped with a map to $\mathrm S_n$ via which it acts on $K^n$?) – LSpice Feb 26 '19 at 15:42
• No, it's not been done. First, the "definition" of inner semidirect product is only worthwhile because of a basic lemma (which says that if $G$ has two subgroups such that blabla then we have an isomorphism of $G$ with the semidirect product blabla). The question is to have such a characterization. In any case, I can't say much since the OP's definition of semidirect product is not standard (one does not allow arbitrary homomorphisms $\Phi:H\to\mathrm{Aut}(K^n)$ in wreath products). – YCor Feb 26 '19 at 16:36
• Your definition of the semidirect product has a typo: the intersection of $K$ and $H$ is the identity, not $H$. And your definition of wreath product is also incorrect. The wreath product requires an action of $H$ on a set $\Omega$, and it is a semidirect product of $K^{\Omega}$ with $H$, and the action of $H$ is by permuting factors, so it’s a very specific type of map into $\mathrm{Aut}(K^{\Omega})$. – Arturo Magidin Mar 2 '19 at 20:50

## 1 Answer

In the case of a transitive permutation action by the complement $$H$$, there is the following characterization, which may answer your question. If $$G$$ has subgroups $$K$$ and $$H$$ such that $$G=\left$$, $$\left$$ is the direct product of all the (distinct) $$H$$-conjugates of $$K$$, $$\left\cap H=1$$, and $$N_H(K)=C_H(K)$$, then $$G\cong K\wr H$$, where the wreath product is relative to the permutation action of $$H$$ on the coset space $$H/N_H(K)$$. It is easy to see, conversely, that a wreath product $$K_1\wr H_1$$ has such subgroups $$K$$ and $$H$$, isomorphic to $$K_1$$ and $$H_1$$ (assuming again a transitive action of $$H_1$$).