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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?

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  • $\begingroup$ 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 $K^n$?) $\endgroup$ – LSpice Feb 26 '19 at 15:42
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    $\begingroup$ 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). $\endgroup$ – YCor Feb 26 '19 at 16:36
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    $\begingroup$ 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 $\mathrm{Aut}(K^{\Omega})$. $\endgroup$ – Arturo Magidin Mar 2 '19 at 20:50
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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 $G=\left<H,K\right>$, $\left<K^H\right>$ is the direct product of all the (distinct) $H$-conjugates of $K$, $\left<K^H\right>\cap H=1$, 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 $H_1$).

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