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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.

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 [

Bamberg [

Tamuli [

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 [

An action of a group G on a non-empty set Ω is a map μ : G × Ω → Ω denoted by μ ( g , α ) = α g for all g ∈ G , α ∈ Ω such that

1) ( α g ) h = α g h for all α ∈ Ω and all g , h ∈ G (1)

2) α 1 = α for all α ∈ Ω (2)

We then say that G acts onΩ.

If G and H are groups, then G × H is a group called the Direct Product of G and H where G × H = { ( g , h ) | g ∈ G , h ∈ H } and multiplication is defined by

( g 1 , h 1 ) ( g 2 , h 2 ) = ( g 1 g 2 , h 1 h 2 ) (3)

If 1 G is the identity for G, and 1 H is the identity for H, then ( 1 G , 1 H ) is the identity for

G × H and ( g , h ) − 1 = ( g − 1 , h − 1 ) (4)

(see details in [

If Γ and Δ are nonempty sets, then we call Γ Δ to denote the set of all functions from Δ to Γ. In the case that C is a group, we turn C Δ into a group by defining product “pointwise”

f g ( γ ) : = f ( γ ) g ( γ ) (5)

for all f , g ∈ C Δ and γ ∈ Δ 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 the semidirect product C Δ ⋊ D = C w r D where D acts on the group C Δ via

f d ( γ ) : = f ( γ d − 1 ) (6)

for all f ∈ C Δ , γ ∈ Δ and d ∈ D and multiplication for all ( f 1 , d 1 ) , ( f 2 , d 2 ) ∈ C w r D is given by

( f 1 , d 1 ) ( f 2 , d 2 ) = ( f 1 f 2 d 1 − 1 , d 1 d 2 ) (7)

Clearly, | C w r D | = | C | | Δ | | D | (8)

(see details in [

A homomorphism ϕ : G → H 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 ϕ : G → H is an embedding, then ker ( ϕ ) = { e G } and from the First Isomorphism Theorem, Im ( ϕ ) ≅ G / { e G } ≅ G . Now Im ( ϕ ) ≤ H as ϕ : G → H is a homomorphism, and so we conclude that in an embedding, G is isomorphic to a subgroup of H. In symbol G ≲ H .

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 [

Proof: Let ψ : G → G Λ be a homomorphism of G onto G Λ with kernel K : = G ( Λ ) . Let ψ be defined by

ψ ( t u ) = u (9)

for each u ∈ G Λ . If x ∈ G , then

ψ ( t u x ) = ψ ( t u ) ψ ( x )

= u ψ ( x ) (by (9))

= ψ ( t u ψ ( x ) ) (by (9))

Therefore, since ψ ( t u x ) = ψ ( t u ψ ( x ) ) it implies that ψ ( t u x ) ψ − 1 ( t u ψ ( x ) ) = 1 . Therefore, ψ ( t u x t u ψ ( x ) − 1 ) = 1 . Thus t u x t u ψ ( x ) − 1 lies in the kernel K. That is, t u x t u ψ ( x ) − 1 ∈ G ( Λ ) . By the definition of wreath product, we can define a function for each x ∈ G , such that f x : G Λ → X Δ by

f x ( u ) : = t u x t u ψ ( x ) − 1 (10)

for all u ∈ G Λ .

We claim that ϕ ( x ) : = ( f x , ψ ( x ) ) ∈ X Δ w r G Δ defines an embedding ϕ of G into X Δ w r G Δ with the function (10).

We seek to show that ϕ is a homomorphism and is injective, hence an embedding.

Take x , y ∈ G , then

ϕ ( x ) ϕ ( x ) = ( f x , ψ ( x ) ) ( f y , ψ ( y ) )

= ( f x f y ψ ( x ) − 1 , ψ ( x ) ψ ( y ) ) (by (7))

Now since ψ is a homomorphism,

ψ ( x ) ψ ( y ) = ψ ( x y ) (11)

By (10), f x f y ψ ( x ) − 1 can be expressed for all u ∈ G Λ as follows:

f x ( u ) f y ψ ( x ) − 1 ( u ) t u ψ ( x y ) = f x ( u ) f y ( u ψ ( x ) ) t u ψ ( x ) ψ ( y ) (by (6) and (11))

= f x ( u ) t u ψ ( x ) t u ψ ( x ) ψ ( y ) − 1 t u ψ ( x ) ψ ( y ) y (by (10))

= f x ( u ) t u ψ ( x ) y

= t u x t u ψ ( x ) − 1 t u ψ ( x ) y (by (10))

= t u x y

= f x y ( u ) t u ψ ( x y ) (by (10))

Therefore, we have that

f x f y ψ ( x ) − 1 = f x y (12)

Hence ϕ ( x ) ϕ ( x ) = ( f x y , ψ ( x y ) ) = ϕ ( x y ) (by (11) and (12)). Hence ϕ is a homomorphism.

Next, we show that ϕ is injective. Now ker ϕ = 1 since ϕ ( x ) = 1 implies f x = 1 and ψ ( x ) = 1 , and so x = t 1 − 1 f x ( 1 ) t 1 ψ ( x ) = t 1 − 1 t 1 = 1 . Thus G ≲ X Δ w r G Δ .

Example 1:

Let G = 〈 ( 12345678 ) 〉 = { ( 1 ) , ( 12345678 ) , ( 1357 ) ( 2468 ) , ( 14725836 ) , ( 15 ) ( 26 ) ( 37 ) ( 48 ) , ( 16385274 ) , ( 1753 ) ( 2864 ) , ( 18765432 ) } is a transitive group. Then Δ = { 1 , 3 , 5 , 7 } is block of G. X = G Δ = { ( 1 ) , ( 1357 ) ( 2468 ) , ( 15 ) ( 26 ) ( 37 ) ( 48 ) , ( 1753 ) ( 2864 ) } . And X Δ = ( 1357 ) = { ( 1 ) , ( 1357 ) , ( 15 ) ( 37 ) , ( 1753 ) } ≅ C 4 , G Δ = ( 1357 ) = { ( 1 ) , ( 1357 ) , ( 15 ) ( 37 ) , ( 1753 ) } ≅ C 4 . Thus X Δ w r G Δ ≅ 〈 ( 1 2 3 4 ) , ( 5 6 7 8 ) , ( 9 10 11 12 ) , ( 13 14 15 16 ) , ( 1 5 9 13 ) ( 2 6 10 14 ) ( 3 7 11 15 ) ( 4 8 12 16 ) 〉 which is a group of order 1024.

Proposition 2: Let G be an arbitrary subgroup with a normal subgroup N and γ : G → G / N be the natural homomorphism. Suppose that ϕ : G → G w r G / N is a homomorphism then there is an embedding ψ : G / N → G w r G / N making the diagram

Proof: If the diagram is to commute (see

ψ ( x N ) = ψ γ ( x ) = ϕ ( x ) (13)

and that is the only way it can be defined. First, we notice that (13) reposes only on the coset xN and not on the representativex. For if, x N = x ′ N , then N = x − 1 x ′ N , and x − 1 x ′ ∈ N .

Hence x − 1 x ′ ∈ ker ϕ and so ϕ ( x − 1 x ′ ) = 1

⇒ ϕ ( x − 1 ) ϕ ( x ′ ) = 1

⇒ ϕ − 1 ( x ) ϕ ( x ′ ) = 1 hence

ϕ ( x ) = ϕ ( x ′ ) and so (13) defines a map.

Next, if we have a different element of G/N, say yN, then

ψ ( x N ⋅ y N ) = ψ ( x y N )

= ϕ ( x y ) (by (13))

= ϕ ( x ) ϕ ( y )

= ψ ( x N ) ψ ( y N ) (by (13))

Thus ψ is a homomorphism. From (13), ψ ( x N ) = 1 ⇒ ϕ ( x ) = 1 ⇒ x ∈ ker ϕ . Thus an embedding. Therefore ker ψ = ker ϕ / N .

Example 2:

Let G = S 3 = 〈 ( 123 ) , ( 12 ) 〉 = { ( 1 ) , ( 12 ) , ( 13 ) , ( 23 ) , ( 123 ) , ( 132 ) } . N = A 3 = 〈 ( 123 ) 〉 = { ( 1 ) , ( 123 ) , ( 132 ) } is a normal subgroup of G. The factor group G / N = { { ( 1 ) , ( 123 ) , ( 132 ) } , { ( 12 ) , ( 13 ) , ( 23 ) } } ≅ C 2 . Then the wreath product

G w r G / N = 〈 ( 123 ) , ( 12 ) , ( 456 ) , ( 45 ) , ( 14 ) ( 25 ) ( 36 ) 〉 = { ( 1 ) , ( 56 ) , ( 45 ) , ( 456 ) , ( 465 ) , ( 46 ) , ( 23 ) , ( 23 ) ( 56 ) , ( 23 ) ( 45 ) , ( 23 ) ( 456 ) , ( 23 ) ( 465 ) , ( 23 ) ( 46 ) , ( 12 ) , ( 12 ) ( 56 ) , ( 12 ) ( 45 ) , ( 12 ) ( 456 ) , ( 12 ) ( 465 ) , ( 12 ) ( 46 ) , ( 123 ) , ( 123 ) ( 56 ) , ( 123 ) ( 45 ) , ( 123 ) ( 456 ) , ( 123 ) ( 465 ) , ( 123 ) ( 46 ) , ( 132 ) , ( 132 ) ( 56 ) , ( 132 ) ( 45 ) , ( 132 ) ( 456 ) , ( 132 ) ( 465 ) , ( 132 ) ( 46 ) , ( 13 ) ,

( 13 ) ( 56 ) , ( 13 ) ( 45 ) , ( 13 ) ( 456 ) , ( 13 ) ( 465 ) , ( 13 ) ( 46 ) , ( 14 ) ( 25 ) ( 36 ) , ( 14 ) ( 2536 ) , ( 1425 ) ( 36 ) , ( 142536 ) , ( 143625 ) , ( 1436 ) ( 25 ) , ( 14 ) ( 2635 ) , ( 14 ) ( 26 ) ( 35 ) , ( 142635 ) , ( 1426 ) ( 35 ) , ( 1435 ) ( 26 ) , ( 143526 ) , ( 1524 ) ( 36 ) , ( 153624 ) , ( 15 ) ( 24 ) ( 36 ) , ( 1536 ) ( 24 ) , ( 15 ) ( 2436 ) , ( 152436 ) , ( 152634 ) ,

( 1534 ) ( 26 ) , ( 15 ) ( 2634 ) , ( 153426 ) , ( 15 ) ( 26 ) ( 34 ) , ( 1526 ) ( 34 ) , ( 163524 ) , ( 1624 ) ( 35 ) , ( 1635 ) ( 24 ) , ( 16 ) ( 24 ) ( 35 ) , ( 162435 ) , ( 16 ) ( 2435 ) , ( 1634 ) ( 25 ) , ( 162534 ) , ( 163425 ) , ( 16 ) ( 2534 ) , ( 1625 ) ( 34 ) , ( 16 ) ( 25 ) ( 34 ) }

which is a group of order 72.

Proposition 3: Let G be an arbitrary group with a normal subgroup N, and put K : = G / N . Then there is an embedding ϕ : G → K w r N such that ϕ maps K onto Im ϕ ∩ B where B is the base group of K w r N . (Thus K w r N contains an isomorphic copy of every extension G of K by N.)

Proof: Let ψ : G → N be the natural homomorphism of G onto K. Let T : = { t u | u ∈ K } be a set of right coset representatives of N in G such that

ψ ( t u ) = u (14)

for each u ∈ K . If x ∈ G ,

ψ ( t u x ) = ψ ( t u ) ψ ( x )

= u ψ ( x ) (by (14))

= ψ ( t u ψ ( x ) ) (by (14))

Therefore, since ψ ( t u x ) = ψ ( t u ψ ( x ) ) it implies that ψ ( t u x ) ψ − 1 ( t u ψ ( x ) ) = 1 . Therefore, ψ ( t u x t u ψ ( x ) − 1 ) = 1 . Thus t u x t u ψ ( x ) − 1 lies in ker ( ψ ) . That is, t u x t u ψ ( x ) − 1 ∈ ker ( ψ ) . By the definition of the wreath product, we can define a function f x for each x ∈ G , such that f x : N → K by

f x ( u ) : = t u x t u ψ ( x ) − 1 (15)

for all u ∈ N .

We claim that ϕ ( x ) : = ( f x , ψ ( x ) ) ∈ K w r N defines an embedding ϕ of G into K w r N with the function (15).

We seek to show that ϕ is an embedding.

Take x , y ∈ G , then

ϕ ( x ) ϕ ( x ) = ( f x , ψ ( x ) ) ( f y , ψ ( y ) )

= ( f x f y ψ ( x ) − 1 , ψ ( x ) ψ ( y ) ) (by (7))

Now since ψ is a homomorphism,

ψ ( x ) ψ ( y ) = ψ ( x y ) (16)

By (15), f x f y ψ ( x ) − 1 can be expressed for all u ∈ N as follows:

f x ( u ) f y ψ ( x ) − 1 ( u ) t u ψ ( x y ) = f x ( u ) f y ( u ψ ( x ) ) t u ψ ( x ) ψ ( y ) (by (15) and (16))

= f x ( u ) t u ψ ( x ) t u ψ ( x ) ψ ( y ) − 1 t u ψ ( x ) ψ ( y ) y (by (15))

= f x ( u ) t u ψ ( x ) y

= t u x t u ψ ( x ) − 1 t u ψ ( x ) y (by (15))

= t u x y

= f x y ( u ) t u ψ ( x y ) (by (15))

Therefore, we have that

f x f y ψ ( x ) − 1 = f x y (17)

Hence ϕ ( x ) ϕ ( x ) = ( f x y , ψ ( x y ) ) = ϕ ( x y ) (by (16) and (17)). Hence ϕ is a homomorphism.

Next, we show that ϕ is injective. Now ker ϕ = 1 since ϕ ( x ) = 1 implies f x = 1 and ψ ( x ) = 1 , and so x = t 1 − 1 f x ( 1 ) t 1 ψ ( x ) = t 1 − 1 t 1 = 1 .

Finally, ϕ ( x ) lies in B when ψ ( x ) = 1 , and this happens exactly when x ∈ N . Thus G ≲ K w r N .

Example 3:

Let G = 〈 ( 123456 ) 〉 = { ( 1 ) , ( 123456 ) , ( 135 ) ( 246 ) , ( 14 ) ( 25 ) ( 36 ) , ( 153 ) ( 264 ) , ( 165432 ) } and N = 〈 ( 135 ) ( 246 ) 〉 = { ( 1 ) , ( 135 ) ( 246 ) , ( 153 ) ( 264 ) } ≅ C 3 is a normal subgroup of G. Then K = G / N = { { ( 1 ) , ( 135 ) ( 246 ) , ( 153 ) ( 264 ) } , { ( 123456 ) , ( 14 ) ( 25 ) ( 36 ) , ( 165432 ) } } ≅ C 2 . Then the wreath product

K w r N ≅ { ( 1 ) , ( 56 ) , ( 34 ) , ( 34 ) ( 56 ) , ( 12 ) , ( 12 ) ( 56 ) , ( 12 ) ( 34 ) , ( 12 ) ( 34 ) ( 56 ) , ( 135 ) ( 246 ) , ( 135246 ) , ( 136245 ) , ( 136 ) ( 245 ) , ( 146235 ) , ( 146 ) ( 235 ) , ( 145 ) ( 236 ) , ( 145236 ) , ( 153 ) ( 264 ) , ( 154263 ) , ( 153264 ) , ( 154 ) ( 263 ) , ( 164253 ) , ( 163 ) ( 254 ) , ( 164 ) ( 253 ) , ( 163254 ) }

which is a group of order 24.

Proposition 4: Let G be any arbitrary group with a normal subgroup N and put K : = G / N . Then there is an embedding ϕ : K w r K → G such that ϕ maps Im ϕ ∩ B onto K, where B is the base group of K w r K and | N | 2 | | G | . (Thus K w r K contains an isomorphic copy of every extension G of K by K.)

Proof: Let ψ : G → K be the natural homomorphism of G onto K with kernel N. Let T : = { t u | u ∈ K } be a set of right coset representatives of N in G such that

ψ ( t u ) = u (18)

for each u ∈ K . If x ∈ G , then

ψ ( t u x ) = ψ ( t u ) ψ ( x )

= u ψ ( x ) (by (18))

= ψ ( t u ψ ( x ) ) (by (18))

Therefore, since ψ ( t u x ) = ψ ( t u ψ ( x ) ) it follows that ψ ( t u x ) ψ − 1 ( t u ψ ( x ) ) = 1 . Therefore, ψ ( t u x t u ψ ( x ) − 1 ) = 1 . Thus t u x t u ψ ( x ) − 1 lies in the kernel N, that is, t u x t u ψ ( x ) − 1 ∈ N . By the definition of wreath product, we can define a function for each x ∈ G , such that f x : K → K by

f x ( u ) : = t u x t u ψ ( x ) − 1 (19)

for all u ∈ K .

Now ϕ ( ( f x , ψ ( x ) ) ) = x ∈ G defines an embedding ϕ of K w r K into G with the function (18).

We seek to show that ϕ is a homomorphism and injective, hence an embedding.

Take x , y ∈ G , then

ϕ ( ( f x , ψ ( x ) ) ) ϕ ( ( f y , ψ ( y ) ) ) = x y = ϕ ( f x y , ψ ( x y ) )

Now since ψ is a homomorphism,

ψ ( x ) ψ ( y ) = ψ ( x y ) (20)

By (17), f x y can be expressed for all u ∈ K as follows:

f x y ( u ) t u ψ ( x y ) = t u x y (by (19))

= t u x t u ψ ( x ) − 1 t u ψ ( x ) y (by (19))

= f x ( u ) t u ψ ( x ) y

= f x ( u ) t u ψ ( x ) t u ψ ( x ) ψ ( y ) − 1 t u ψ ( x ) ψ ( y ) y (by (19))

= f x ( u ) f y ( u ψ ( x ) ) t u ψ ( x ) ψ ( y ) (by (6) and (20))

= f x ( u ) f y ψ ( x ) − 1 ( u ) t u ψ ( x y ) (by (6) and (20))

Therefore, we have

f x y = f x f y ψ ( x ) − 1 (21)

Hence ϕ ( ( f x , ψ ( x ) ) ) ϕ ( ( f y , ψ ( y ) ) ) = ( f x f y ψ ( x ) − 1 , ψ ( x ) ψ ( y ) ) = ϕ ( ( f x , ψ ( x ) ) ( f y , ψ ( y ) ) ) (by (20) and (21)), showing that ϕ is a homomorphism.

Next, we show that ϕ is injective. Now ker ϕ = 1 since ϕ ( ( f x , ψ ( x ) ) ) = 1 implies x = 1 and so f 1 = 1 and ψ ( 1 ) = 1 , thus from Lagrange theorem, | G | | K | | K | | K | = | N | 2 | G | , i.e. | N | 2 | | G | as f 1 = 1 and ψ ( 1 ) = 1 .

Finally, ϕ ( ( f x , ψ ( x ) ) ) lies in B when ψ ( x ) = 1 , and this happens exactly when x ∈ N .

Thus K w r K ≲ G .

Example 4:

Let G = S 4 = { ( 1 ) , ( 34 ) , ( 23 ) , ( 234 ) , ( 243 ) , ( 24 ) , ( 12 ) , ( 12 ) ( 34 ) , ( 123 ) , ( 1234 ) , ( 1243 ) , ( 124 ) , ( 132 ) , ( 1342 ) , ( 13 ) , ( 134 ) , ( 13 ) ( 24 ) , ( 1324 ) , ( 1432 ) , ( 142 ) , ( 143 ) , ( 14 ) , ( 1423 ) , ( 14 ) ( 23 ) } and N = A 4 = { ( 1 ) , ( 234 ) , ( 243 ) , ( 12 ) ( 34 ) , ( 123 ) , ( 124 ) , ( 132 ) , ( 134 ) , ( 13 ) ( 24 ) , ( 142 ) , ( 143 ) , ( 14 ) ( 23 ) } . Then K = G / N = S 4 / A 4 = { { ( 1 ) , ( 234 ) , ( 243 ) , ( 12 ) ( 34 ) , ( 123 ) , ( 124 ) , ( 132 ) , ( 134 ) , ( 13 ) ( 24 ) , ( 142 ) , ( 143 ) , ( 14 ) ( 23 ) } , { ( 34 ) , ( 23 ) , ( 24 ) , ( 12 ) , ( 1234 ) , ( 1243 ) , ( 1342 ) , ( 13 ) , ( 1324 ) , ( 1432 ) , ( 14 ) , ( 1423 ) } } ≅ C 2 . Now | N | 2 | G | = 12 2 24 = 6 . Then the wreath product K w r K ≅ C 2 w r C 2 = { ( 1 ) , ( 34 ) , ( 12 ) , ( 12 ) ( 34 ) , ( 13 ) ( 24 ) , ( 1324 ) , ( 1423 ) , ( 14 ) ( 23 ) } which is a group of order 8.

Remark 5: If ϕ : G → H is an embedding, then n | G | = | H | where n ≠ 1 ∈ ℕ is the index of G by H. If the index n = 1 , then ϕ is as isomorphism.

Theorem 6 (Dixon & Mortimer, 1996): (Universal Embedding Theorem)

Let G be an arbitrary group with a normal subgroup N, and put K : = G / N . Then there is an embedding ϕ : G → N w r K such that ϕ maps N onto Im ϕ ∩ B where B is the base group of N w r K . (Thus N w r K contains an isomorphic copy of every extension G of N by K.)

Condition 1: If G is of order p, i.e. | G | = p , then | N | = p or 1, by Lagrange’s theorem, then either G = N or N = { e } .

If | N | = p , then | G | = | N | = p , then G = N and so K = G / N = G / G ≅ { e } . Thus | K | = 1 , and ϕ being a homomorphism implies that n | G | = | N | | K | | K | and so n = 1 .

If | N | = 1 , then N = { e } and so K = G / N = G / { e } ≅ G . Thus | K | = | G | = p , and ϕ being a homomorphism implies that n | G | = | N | | K | | K | and so n = 1 .

Thus ϕ is an isomorphism if | G | = p or rather ϕ is an isomorphism if G is cyclic.

Condition 2: If the order of G is p 2 , then G is known to be an Abelian group and by Lagrange’s theorem | N | = p 2 ,p or 1.

If | N | = p 2 , then G = N and so | K | = | G / N | = | G | / | N | = p 2 / p 2 = 1 . Thus K = { e } . And ϕ being a homomorphism implies that n | G | = | N | | K | | K | and so n p 2 = p 2 and n = 1 . Therefore, ϕ is an isomorphism if G = N .

If | N | = p , then | K | = | G / N | = | G | / | N | = p 2 / p = p . Thus ϕ being a homomorphism implies that n | G | = | N | | K | | K | and so n p 2 = p p p and n = p p − 1 the index.

If | N | = 1 , then N = { e } and so | K | = | G / N | = | G | / | N | = p 2 / 1 = p 2 . Thus ϕ being a homomorphism implies that n | G | = | N | | K | | K | and so n p 2 = 1 p 2 p 2 = p 2 . Thus n = 1 . Therefore, ϕ is an isomorphism if N = { e } .

Condition 3: If G is a simple group, then either N = G or N = { e } .

If N = G , then | N | = | G | , then | K | = | G | / | N | = 1 , and ϕ being a homomorphism implies that n | G | = | N | | K | | K | and so n = 1 .

If N = { e } , then | N | = 1 , then | K | = | G | / 1 = | G | , and ϕ being a homomorphism implies that n | G | = | N | | K | | K | and so n = 1 .

Thus ϕ is an isomorphism if G is simple.

Example 5:

Let G = 〈 ( 12345 ) 〉 = { ( 1 ) , ( 12345 ) , ( 13524 ) , ( 14253 ) , ( 15432 ) } . Then the normal subgroup N = G or N = { ( 1 ) } . If N = G , then K = G / N = G / G = { { ( 1 ) , ( 12345 ) , ( 13524 ) , ( 14253 ) , ( 15432 ) } } ≅ { ( 1 ) } . Thus N w r K ≅ { ( 1 ) , ( 12345 ) , ( 13524 ) , ( 14253 ) , ( 15432 ) } = G . If N = { ( 1 ) } , then K = G / N = G / { ( 1 ) } = { ( 1 ) , ( 12345 ) , ( 13524 ) , ( 14253 ) , ( 15432 ) } = G . Thus N w r K ≅ { ( 1 ) , ( 12345 ) , ( 13524 ) , ( 14253 ) , ( 15432 ) } = G . Thus an isomorphism as G is Cyclic, simple and its order is prime.

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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Suleiman, E. and Audu, M.S. (2021) On Some Embedment of Groups into Wreath Products. Advances in Pure Mathematics, 11, 109-120. https://doi.org/10.4236/apm.2021.112007