On Some Embedment of Groups into Wreath Products

In this paper, we showed how groups are embedded into wreath products, we gave a simpler proof of the theorem by Audu (1991) (see [1]), also proved that a group can be embedded into the wreath product of a factor group by a normal subgroup and also proved that a factor group can be embedded inside a wreath product and the wreath product of a factor group by a factor group can be embedded into a group. We further showed that when the abstract group in the Universal Embedding Theorem is a p-group, cyclic and simple, the embedding becomes an isomorphism. Examples were given to justify the results.


Introduction
Embedding is one of the most important properties of wreath product; this property was further investigated with regards to imprimitivity of groups, normal subgroups and Quotient Group. Many people have worked on wreath products over the years and their work is as shown below: Suzuki [2] in 1982 proved the Kaloujnine-Krasner Theorem that states that if F is a group extension of N by G, then F can be embedded inside the standard wreath product N Wr G . Audu [1] in 1991 proved that a permutation group that is transitive and imprimitive that is acting on a finite set can be embedded inside the wreath product . They showed that the group which is imprimitive can be embedded inside the wreath product Bamberg [5] in 2005 states that if G is any transitive imprimitive permutation group on a set Ω and Λ a G-invariant partition of Ω, if also Δ is an element of Λ and C the permutation group induced by the action of { } Δ G on Δ. If D is the group of permutations induced by G on Λ, then Ω may be identified with Δ Λ × in such a way that G can be embedded into the wreath product C wr D in imprimitive action. Bamberg further states that if G is transitive but imprimitive group on a finite set Ω, then G can be embedded into the wreath product G embeds into a wreath product. As Ω is finite, the process can continue until an embedding of G into iterated wreath product of primitive groups was found. Chan [6] in 2006 proved that every faithful group action that is transitive and imprimitive is embeddable in a wreath product. Cameron [7] in 2013 showed that if H is a permutation group induced on a part by its setwise stabilizer and if K is the permutation group induced on the set of parts by the group G, then G is embedded in the wreath product H wr K .
Tamuli [8]  φ  , where B is the base group of N wr K . Mikaelian [9] in 2002 showed that every extension of a group G where the group product is the product variety that consists of all extensions of groups, if N is a normal subgroup and , then every extension of G can be isomorphically embedded into the wreath product N Wr H . Hulpke [10] in 2004 proved that a transitive group G Given isomorphism between two groups, knowing how the first group is isomorphic to a subgroup of the other groups helps us to know the structures being preserved. Since a wreath product is a group with many subgroups, it is easily seen that to be isomorphic to a group.
In this paper, we were able to give new proof of the theorem by Audu (1991) (see [1]); we obtained the proof of the following: a group can be embedded into the wreath product of a factor group by a normal subgroup; the wreath product of two factor groups can be embedded into a group; when the abstract group in the Universal Embedding Theorem is a p-group, cyclic and simple, the embedding is an isomorphism.

Basic Definitions
An action of a group G on a non-empty set Ω is a map : G µ × Ω → Ω denoted by ( ) for all α ∈ Ω and all , (see details in [11]) If Γ and Δ are nonempty sets, then we call Δ Γ to denote the set of all functions from Δ to Γ. In the case that C is a group, we turn Δ C into a group by defining product "pointwise" for all Δ , f g C ∈ and Δ γ ∈ where the product in the right is in C.
Let C and D be groups and suppose D acts on the nonempty set Δ. Then the wreath product of C by D is defined with respect to this action is defined to be for all (see details in [12]) that is one-to-one (injective) is called an embedding: the group G "embeds" into H as a subgroup. If φ is not one-to-one, then it is a quotient. Note that if and from the First Isomorphism Theorem, is a homomorphism, and so we conclude that in an embedding, G is isomorphic to a subgroup of H. In symbol G H  .

Results
We now give an alternative proof to a theorem of Audu (1991) and also outline some propositions with their proofs. We proved embedding by showing that they are homomorphic and injective. We gave three conditions on the Universal Embedding Theorem (Dixon & Mortimer, 1996) when the group is a p-group and when the group is simple.
Theorem 1 (see [1]): Let G be any transitive and imprimitivity group acting on a set Ω; let ≤ ≤ be a system of imprimitivity of G and Δ be an Therefore, By the definition of wreath product, we can define a function for We claim that ( ) ( ) : , We seek to show that φ is a homomorphism and is injective, hence an embedding.
Next, we show that φ is injective. Now ker Example 1:  Proof: If the diagram is to commute (see Figure 1), then we have for an arbi- and that is the only way it can be defined. First, we notice that (13) and so (13) defines a map.
Next, if we have a different element of G/N, say yN, then . Thus an embedding. Therefore ker ker N ψ φ = . Therefore, By the definition of the wreath product, we can define a function x f for each x G ∈ , such that : We claim that ( ) ( )

( )
: , defines an embedding φ of G into K wr N with the function (15).
We seek to show that φ is an embedding.
Next, we show that φ is injective. Now ker  Proposition 4: Let G be any arbitrary group with a normal subgroup N and put : Then there is an embedding : By the definition of wreath product, we can define a function for each x G ∈ , such that : defines an embedding φ of K wr K into G with the function (18).
We seek to show that φ is a homomorphism and injective, hence an embedding.
Next, we show that φ is injective. Now ker   , and φ being a homomorphism implies that K n G N K = and so 1 n = .
Thus φ is an isomorphism if G is simple.

Discussion
Embedding is an important property of wreath product as it helps in preserving structures between groups. Under some conditions we have seen that the Universal embedding Theorem is an isomorphism.

Conclusion
In this paper, we were able to give a new proof of the theorem by Audu (1991), which proved that a group can be embedded into the wreath product of a factor group by a normal subgroup and also proved that a factor group can be embedded inside a wreath product and the wreath product of a factor group by a factor group can be embedded into a group. It was shown that when the abstract group in the universal embedding theorem is a p-group, cyclic and simple, the embedding is an isomorphism.