Let H and A be groups, let K = A^H be the set of all functions from H to A, and consider the action of H on itself by right multiplication. This action extends naturally to an action of H on K defined by ${\displaystyle \phi (g).h=\phi (gh^{-1}),}$  where ${\displaystyle \phi \in K,}$  and g and h are both in H. This is an automorphism of K, so we can define the semidirect product K ⋊ H called the regular wreath product, and denoted A Wr H or ${\displaystyle A\wr H.}$  The group K = A^H (which is isomorphic to ${\displaystyle \{(f_{x},1)\in A\wr H:x\in K\}}$ ) is called the base group of the wreath product.

The Krasner–Kaloujnine universal embedding theorem states that if G has a normal subgroup A and H = G/A, then there is an injective homomorphism of groups ${\displaystyle \theta :G\to A\wr H}$  such that A maps surjectively onto ${\displaystyle {\text{im}}(\theta )\cap K.}$ ^[2] This is equivalent to the wreath product A Wr H having a subgroup isomorphic to G, where G is any extension of H by A.

This proof comes from Dixon–Mortimer.^[3]

Define a homomorphism ${\displaystyle \psi :G\to H}$  whose kernel is A. Choose a set ${\displaystyle T=\{t_{u}:u\in H\}}$  of (right) coset representatives of A in G, where ${\displaystyle \psi (t_{u})=u.}$  Then for all x in G, ${\displaystyle t_{u}xt_{u\psi (x)}^{-1}\in \ker \psi =A.}$  For each x in G, we define a function f_x: H → A such that ${\displaystyle f_{x}(u)=t_{u}xt_{u\psi (x)}^{-1}.}$  Then the embedding ${\displaystyle \theta }$  is given by ${\displaystyle \theta (x)=(f_{x},\psi (x))\in A\wr H.}$ 

We now prove that this is a homomorphism. If x and y are in G, then ${\displaystyle \theta (x)\theta (y)=(f_{x}(f_{y}.\psi (x)^{-1}),\psi (xy)).}$  Now ${\displaystyle f_{y}(u).\psi (x)^{-1}=f_{y}(u\psi (x)),}$  so for all u in H,

${\displaystyle f_{x}(u)(f_{y}(u).\psi (x))=t_{u}xt_{u\psi (x)}^{-1}t_{u\psi (x)}yt_{u\psi (x)\psi (y)}^{-1}=t_{u}xyt_{u\psi (xy)}^{-1},}$ 

so f_x f_y = f_xy. Hence ${\displaystyle \theta }$  is a homomorphism as required.

The homomorphism is injective. If ${\displaystyle \theta (x)=\theta (y),}$  then both f_x(u) = f_y(u) (for all u) and ${\displaystyle \psi (x)=\psi (y).}$  Then ${\displaystyle t_{u}xt_{u\psi (x)}^{-1}=t_{u}yt_{u\psi (y)}^{-1},}$  but we can cancel t_u and ${\displaystyle t_{u\psi (x)}^{-1}=t_{u\psi (y)}^{-1}}$  from both sides, so x = y, hence ${\displaystyle \theta }$  is injective. Finally, ${\displaystyle \theta (x)\in K}$  precisely when ${\displaystyle \psi (x)=1,}$  in other words when ${\displaystyle x\in A}$  (as ${\displaystyle A=\ker \psi }$ ).

• The Krohn–Rhodes theorem is a statement similar to the universal embedding theorem, but for semigroups. A semigroup S is a divisor of a semigroup T if it is the image of a subsemigroup of T under a homomorphism. The theorem states that every finite semigroup S is a divisor of a finite alternating wreath product of finite simple groups (each of which is a divisor of S) and finite aperiodic semigroups.
• An alternate version of the theorem exists which requires only a group G and a subgroup A (not necessarily normal).^[4] In this case, G is isomorphic to a subgroup of the regular wreath product A Wr (G/Core(A)).

• Dixon, John; Mortimer, Brian (1996). Permutation Groups. Springer. ISBN 978-0387945996.
• Kaloujnine, Lev; Krasner, Marc (1951a). "Produit complet des groupes de permutations et le problème d'extension de groupes II". Acta Sci. Math. Szeged. 14: 39–66.
• Kaloujnine, Lev; Krasner, Marc (1951b). "Produit complet des groupes de permutations et le problème d'extension de groupes III". Acta Sci. Math. Szeged. 14: 69–82.
• Praeger, Cheryl; Schneider, Csaba (2018). Permutation groups and Cartesian Decompositions. Cambridge University Press. ISBN 978-0521675062.