The Wreath Product of Powerful p-Groups

Abstract

This study provides a scholarly examination of fundamental concepts within the field of group theory, specifically focusing on topics such as the wreath product and powerful p-groups. We examine the characteristics pertaining to the structure of the wreath product of cyclic p-groups, with a specific focus on the groups that are powerfully embedded within it. The primary discovery pertains to the construction of the powerful wreath product and the quasi-powerful wreath product. In this study, we establish that subgroups are powerful within the wreath product, specifically focusing on p-groups. The aforementioned outcome is derived from the assumption that p is a prime number and W is the standard wreath product of two nontrivial cyclic p-groups, denoted as G and H.

1. Introduction

The field of group theory encompasses a wide range of topics and areas of study. Currently, we are dealing with powerful p-groups and wreath products, two of the most important tools in group theory. The main goal of this paper is to investigate the conditions under which the order of the wreath product of two nontrivial cyclic p-groups must be considered finite. So, unless otherwise stated, we assume that all cyclic p-groups mentioned are finite and that all sets are nonempty finite sets.

We look at fundamental facts in the field of abstract algebra. Let p be a prime number. Let G be a nontrivial p-group. We defined a type of finite p-groups known as powerful wreath products. The classification of finite p-groups remains a significant issue in group theory. It would be acceptable to expect that these groups could be usefully classified. Throughout this work, p represents a prime number.

This paper is organized as follows. This section contains three subsections, including a literature review related to work, recall, and proving the main concepts of a powerful p-group and wreath product. Section 2, includes methods that we have used to study this work. Section 3 contains the principal results and is divided into two subsections. In the first subsection, we provide evidence that supports the claim that specific subgroups of the wreath product possess the defining characteristics of a powerful p-group. The existence of instances of the influential subgroup within the wreath product can be observed. Lemma 9 demonstrates that groups that construct the wreath product exhibit the characteristic of being powerful p-groups. The center of the base group of a wreath product is a powerful p-group. The Lemma 10 demonstrates that the commutator subgroup of a wreath product possesses the property of being powerful. In the second, one of the motivations behind this work is to introduce new classes of potent p-groups known as powerful wreath products and quasi-powerful wreath products. Our primary focus revolves around the establishment of essential and comprehensive criteria that determine whether a wreath product possesses the attributes of being powerful and quasi-powerful. The aforementioned result is based on the assumption that p is a prime number and W is the standard wreath product of two nontrivial cyclic p-groups, denoted as G and H. Lemma 12 says the wreath product of two permutation cyclic p-groups is called a quasi-powerful p-group. Also, Lemma 13 shows that under certain conditions, the wreath product must be powerful and Lemma 18 shows that W is the ${\mu }_i$ group of exponent p. In addition, Lemma 17 assumes that G and H are finite cyclic p-groups that act on finite sets X and Y, respectively, such that there exists a homomorphism function from G into H. Let $W=G\wr H$ be a group act on $X\times Y$. If $ker\theta$ is powerful and B is powerfully embedded in $W,$ then $(W/ker\theta )\wr H$ is powerful. Finally, if B and p-subgroup H are powerfully embedded, we can draw two conclusions. First, it gives us the correlation between a subgroup of wreath products, a subgroup of base wreath products, and a subgroup of group-constructed wreath products such that all subgroups are generated by all $p^i$th powers, $\forall i\in \mathbb{N}$. The second result relates the Frattini subgroup of the base group that forms the wreath product to the Frattini subgroup of the wreath product. In Section 4 and Section 5, we cover this work with a discussion and conclusions.

1.1. Literature Review

In 1933, The work of Hall in [1] created interest in the study of power structures in finite p-groups. Since then, many families of p-groups have been shown to have a regular power structure, and even more have been shown to have some subset of these properties. N. Blackburn determined the p-groups of rank 2. In 1961, his results implied that the maximal class 3-groups and groups of order $p^3$ and $p^4$ are not powerful, but they contain a powerful maximal subgroup [2]. In 1973, the author of [3] presented minimal generating sets for some wreath products of groups. The result of Gaschutz shows that if G is a finite nilpotent group, if H is a finite non-trivial group, and $g.c.d\left(\right|G|,|H\left|\right)=1,$ then $d(G\wr H)=max\{1+d(G),d(H\left)\right\}$. In 1969, Arganbright [4] began studying what are now known as potent p-groups. In 1987, this family of groups was introduced in the paper [5] by Lubotzky and Mann. They defined that a powerful p-group is one whose subgroup generated by all pth powers contains the commutator subgroup. In addition, let N be a normal subgroup of G. Then, N is powerfully embedded in G, if the subgroup of N that is generated by all pth powers contains the commutator of N and G. The concept of powerful p-groups is important. If N is powerfully embedded in G, then it is powerful and, moreover, if $N\le K\le G$ and $K/N$ is cyclic, then K is powerful. Consequently, one may inquire whether a finite p-group G possesses any of the aforementioned desirable characteristics if all of its maximal subgroups are powerful. This naturally leads to the generalization that all subgroups of index $p^i$ are powerful. In 2002, L. Wilson [6] demonstrated that if G is a powerful p-group with p odd, the exponent of ${\mathrm{\Omega }}_i\left(G\right)$ is at most $p^i$ where $i\in \mathbb{N}.$ By applying a recent result of H´ethelyi and L´evai, they proved that $|G^{p^i}|=|G:{\mathrm{\Omega }}_i\left(G\right)|.$ In 2007, the report [7] covers the fundamental methodology for calculating the wreath product of groups. The discussion also encompassed the examination of the circumstances in which the wreath products of permutation groups are faithful, transitive, and primitive. Additionally, an examination was conducted on the central properties of both the stabilizer and the wreath products. Lastly, a visual representation was provided to substantiate the study outcomes. In 2014, the book [8] is structured into three primary chapters: a comprehensive exposition of general theory, an in-depth exploration of wreath products of finite groups, and a detailed analysis of harmonic phenomena within finite wreath products. The properties of wreath products and their general theory are covered in Chapter 2. In 2021, the author of [9] presented the structure of finite p-groups, with the property that every subgroup of index $p^i$ is powerful for some i. He showed that for odd prime p, these groups must be potent under certain conditions. The concept of a quasi-powerful p-group was introduced by the author of [10], specifically focusing on odd primes p.

The book [11] provides a comprehensive overview of the current state of knowledge regarding the proofs of significant structure theorems for p-groups. The notation used in this work is based on some of that from [11,12].

1.2. Powerful p-Group

We discuss the powerful finite p-groups that are described in [5]. In this section, the variable p is used to represent an odd prime number. Unless explicitly specified, let G denote a finite p-group. Let N be a normal subgroup of the group G.

Definition 1.

The commutator of an ordered pair of elements $a,a^{\prime }$ of a group G is the element

$$[a,a^{\prime }]=a^{-1}{a}^{\prime -1}a{a}^{\prime }\in G.$$

Definition 2.

Let N be a subgroup of the group G. Then, the corresponding commutator subgroup is

$$[N,G]=\langle [n,a]:n\in N,a\in G\rangle ⩽G.$$

We emphasize that $[N,G]$ is the subgroup generated by all the commutators $[n,a]$ with $n\in N$ and $a\in G.$ The particular, subgroup $[G,G],$ generated by all commutators in $G,$ is usually denoted by $G^{\prime }$ and called the commutator subgroup of $G.$ In reference [13], several fundamental properties of commutators are detailed.

Definition 3.

(1) 

The group G is said to be powerful if $G/G^{p^i}$ is abelian, $i\in \mathbb{N}$.

(2) 

A normal group N of a finite group G is said to be powerfully embedded in G, if $N^p⩾[N,G].$

Definition 4.

Consider a prime number p such that p is odd. A finite p-group is referred to as quasi-powerful if its quotient group $G/Z\left(G\right)$ is a powerful p-group.

Remark 1.

(1) 

A group G is powerful if and only if $\mathrm{\Theta }\left(G\right)=G^p$, where $\mathrm{\Theta }\left(G\right)$ is a Frattini subgroup of $G.$

(2) 

Any powerfully embedded subgroup is powerful as $[N,N]⩽[N,G]⩽N^p$. Being powerfully embedded in a larger subgroup is a stronger condition than being powerful in this sense.

(3) 

The property of being powerful is preserved by quotients, as stated in Lemma 2.2 (i) in [14]. However, this property is not preserved when considering subgroups. In addition, the direct product of powerful groups is powerful.

In this study, the authors frequently utilize various properties of powerful p-groups, as evidenced by References [5,11], often without explicitly acknowledging them.

Theorem 1.

Let N be powerfully embedded in G. Then, $[N,G]$ and $N^p$ are powerfully embedded in G.

Proof. 

See Theorem 1.1 in [5]. □

Corollary 1.

Let G be powerful. Then, the subgroups $G^{p^i}$ and $\mathrm{\Theta }\left(G\right)$ are powerfully embedded, $i\in \mathbb{N}.$

Proof. 

See Corollary 1.2 in [5]. □

Lemma 1.

If N and M are powerfully embedded in $G.$ Then, $G^{p^i}=N^{p^i}{M}^{p^i},i\in \mathbb{N}$ and

$${\left([N,M]\right)}^{p^i}=\prod _{n=0}^i[N^{p^i},M^{p^i}].$$

(1)

Proof. 

See Proposition 1.6 in [5]. □

Lemma 2.

Consider that N is a powerfully embedded subgroup of $G,$ then the subgroup $K=\langle a\rangle N$ is a powerful p-group, for all $a\in G$ and $[K,K]\le N^p$.

Proof. 

See Lemma 11.7 in [11]. □

Lemma 3.

Let G be a powerful p-group with p odd and $k,j\in \mathbb{N}$. Then, the following statements hold:

(i) 

$[G^{p^k},G]\le G^{p^{k+1}}.$

(ii) 

${\left(G^{p^k}\right)}^{G^j}$.

Proof. 

See Theorem 11.10 in [11]. □

Definition 5.

Let G denote a finite p-group. We define a group G to be a ${\mu }_i$ group if every subgroup of G with index $p^i$ is powerful, where $i\in \mathbb{N}$.

Definition 6.

A finite p-group G has a regular power structure if the following three conditions hold for all positive i integers:

$$G^{p^i}=\{a^{p^i}:a\in G\}.$$

(2)

$${\mathrm{\Omega }}_i\left(G\right)=\{a\in G:|a|\le p^i\}.$$

(3)

$$|G^{p^i}|=|G:{\mathrm{\Omega }}_i\left(G\right)|.$$

(4)

Theorem 2.

Let p be an odd prime. The powerful p-groups have a regular power structure.

The first power structure condition was established by Proposition 1.7 in [5]. The establishment of the second and third power structure conditions can be traced back to the work of Wilson in [6], Theorem 3.1. The proof of the third condition relies on a result obtained by H’ethelyi and L’evai in [15], Corollary 2. Fernández-Alcober presented a more concise demonstration of conditions 3 and 4 as outlined in the work by Fernández-Alcober [16]. This can be observed in Theorem 1 (iii) and Theorem 4. It is worth mentioning that Mazur provided an independent proof of the theorem in [17], Theorem 1, which establishes that $|G^{p^i}|=|G:{\mathrm{\Omega }}_i\left(G\right)|$ holds true for a powerful p-group G.

A generalization of powerful p-groups is the notion of a potent p-group.

Definition 7.

Suppose that p is an odd prime number. A finite p-group is considered potent if ${\gamma }_{p-1}\left(G\right)⩽G^p.$

Theorem 3.

Let G be a p-group and $p>2$. Then, G is metacyclic if $|W:W^p|\le p^2$ holds. The regular p-groups with $d\left(G\right)=2$ are therefore the metracyclic p-groups for $p>2.$

Proof. 

See Theorem 11.4 in [18]. □

1.3. Wreath Product

We collected here the basic definitions and results for a wreath product that will be used in this work. Consider two nontrivial cyclic p-groups, denoted as G and H. Let us assume that H performs an action on a finite set Y. Let $B=\underset{indexbyY}{\overset{G\times G\times \cdots \times G}{︸}}$. An element of the group B is written as a function from the set Y to the group $G.$ The wreath product of G by H according to the action of H on B is denoted by $W=G\wr H$. The base group of the wreath product, B, and H can both be considered subgroups of W. B becomes a normal subgroup of $W.$ In this paper, we discuss the wreath product $C_p\wr C_p$ of two groups of order p such that it is isomorphic to the Sylow p-subgroup of symmetric group on a set of $p^2$ elements. It is exercise 529 (ii) in Reference [19]. The order of the wreath product is $\left|W\right|=\left|H\right|·{\left|G\right|}^{\left|Y\right|}=p^{p+1},$ where G and H are finite p-groups. The identity element of the wreath product W is $e_W=e_B{e}_H$ where $e_B:y⟼e_G.$ In addition, to obtain an inverse element of any element in the wreath product W, we define $g^{-1}\in B$ by $y{g}^{-1}={\left(yg\right)}^{-1}$ then for $w=gh\in W$ the inverse of w is

$${\left(gh\right)}^{-1}={\left(g^{-1}\right)}^h{h}^{-1},$$

where ${\left(g^{-1}\right)}^h={\left(g^h\right)}^{-1}.$ More details are in [20].

Proposition 1.

The wreath product of G and H according to the action of H on B.

(i) 

It is the semidirect product of groups $B⋊H.$

(ii) 

The subgroup B is a normal of W, and

(iii) 

the intersection, $B\cap H=\left\{e\right\}.$

Proof. 

See Proposition 4.4 in [21]. □

Definition 8.

Let W be a wreath product group with identity element $e_W,$ a subgroup H and normal subgroup $B⊲W$. The wreath product W is a product of subgroups, $W=BH,$ and subgroups have trivial intersection.

Definition 9.

A non-abelian spacial group is characterized by having a centralizer that is equal to a prime number. The given group is an extra-special p-group.

Remark 2.

The base group B of the wreath product is isomorphic to extra-special p-group of order $p^p$ and $Z\left(B\right)=[B,B]$. This has order p, so it is cyclic.

The lemma that follows is Theorem 3, which is mentioned in [22].

Lemma 4.

Consider two nontrivial cyclic p-groups, denoted as G and H. Let us assume that H performs an action on a finite set Y. Let wreath product $W=G\wr H$ and $B=G^Y$, then $W/B\cong H.$

Proof. 

This is proven through the application of the Third Theorem on Homomorphisms. □

Definition 10.

A wreath product commutator is denoted by

$$[W,W]=[BH,BH]=\{{\langle gh\rangle }^{-1}{\langle fb\rangle }^{-1}\langle gh\rangle \langle fb\rangle :g,f\in B,h,b\in H\}.$$

A subgroup of wreath product generated by all p-th powers is denoted by $W^p$.

The exponent of a finite group plays an important role in the structure of the group. For example, a finite non-abelian p-group of order $p^3$ has either exponent p or $p^2$. In general, any extra-special p-group has either exponent p or $p^2$. See the book [23] for more details.

Remark 3.

For all positive integers i: $W^{p^i}=\{w^{p^i}:w\in W\},B^{p^i}=\{b^{p^i}:b\in B\}$ and $H^{p^i}=\{h^{p^i}:h\in H\}.$

Definition 11.

The Frattini subgroup of a finite wreath product $W=G\wr H$ is the intersection of all maximal subgroups, denoted by $\mathrm{\Theta }\left(W\right)$. Obviously, if W is a finite wreath product of order $\left|W\right|=p^{n}n\in \mathbb{N},$ then $\mathrm{\Theta }\left(W\right)⊲W$.

In this discussion, we focus on the finite powerful p-groups as outlined in Reference [5]. We proceed to examine their definitions and theorems in the context of wreath products. Currently, we are considering the concepts:

Definition 12.

The wreath product $W=G\wr H$ is considered powerful when the quotient group $W/W^p$ has abelian properties.

Definition 13.

A group H of a finite wreath product of cyclic p-groups is said to be powerfully embedded in $W=G\wr H$, if $H^p⩾[H,W],$ where the commutator subgroup of a subgroup H and a wreath product W is denoted as $[H,W]$ and consists of all elements of the form $h^{-1}{w}^{-1}hw$ such that h is in H and w is in W.

Remark 4.

A group $W=G\wr H$ is powerful if and only if $\mathrm{\Theta }\left(W\right)=W^p$.

Lemma 5.

Consider that B is a powerfully embedded subgroup of $W,$ then the subgroup $K=\langle w\rangle B$ is a powerful p-group, for all $w\in W$ and $[K,K]\le B^p$.

Proof. 

We are dealing with wreath products as a group, applying Lemma 2. We have a requirement. □

Lemma 6.

Let W be a powerful p-group with p odd and $k,j\in \mathbb{N}$. Then, the following statements hold:

(i) 

$[W^{p^k},W]\le W^{p^{k+1}}.$

(ii) 

${\left(W^{p^k}\right)}^{p^j}$.

Proof. 

We are dealing with wreath products as a group, applying Lemma 3. We have a requirement. □

Lemma 7.

Let $(G,X)$ and $(H,Y)$ be two permutation groups, and let A be a subgroup of H. Then, a wreath product $G\wr A$ is a subgroup of a wreath product $W=G\wr H$ using a natural identification of an isomorphic group.

Proof. 

See Lemma 1.9 in [24]. □

Lemma 8.

Let $(G,X)$ and $(H,Y)$ be two permutation groups and K be a subgroup of G. Then, $K\wr H$ can be embedded in a natural way in $G\wr H.$

Proof. 

See Lemma 1.13 in [24]. □

2. Methodology

This section pertains to the methodologies employed in investigating the aforementioned issue. The utilization of relationships between facts in the field of group theory can be employed. The strategies employed are of a conventional nature. The primary instrument employed in this study is the wreath product, which is formed by combining two nontrivial cyclic p-groups and a powerful p-group, where p is a prime number. The theory of p-groups is a significant tool in the study of wreath products. In this study, we shall review established definitions and theorems pertaining to the concept of powerful p-group, and subsequently proceed to analyze their application in the context of the wreath product. In our research, we place significant emphasis on the necessity of acknowledging both the base group and the group that exhibits behavior resembling symmetric groups.

3. Main Results

This section will provide an analysis of the primary findings derived from our research, which will be presented in two distinct subsections. The initial subsection centers its attention on the influential subgroups known as powerful p-groups of wreath products. The primary objective of the second subsection is to establish the necessary and sufficient criteria for a wreath product to exhibit the characteristics of being a powerful and quasi-powerful p-group. Let p represent a prime number.

3.1. Powerful Subgroup of Wreath Product

In this section, we present proof demonstrating that certain subgroups of the wreath product exhibit the characteristic properties of a powerful p-group. The presence of instances of the influential subgroup within the wreath product can be observed. Throughout this study, it is consistently observed that if a group is abelian, it is powerful.

The following lemma shows that groups that construct wreath products are powerful p-groups.

Lemma 9.

For p is an odd prime. Consider wreath product $W=G\wr H$ of cyclic p-groups. Let B be base group of the wreath product. Then, the following statements hold:

(i) 

The group G is a powerful p-group.

(ii) 

The group H is a powerful p-group. Therefore, $W/B$ is a powerful p-group.

Proof. 

Since G and H are cyclic, then G and H are abelian groups. Hence, G and H are powerful. Suppose that p is an odd prime. As in Lemma 4, we find $W/B$ is a powerful p-group. For this result, the condition of powerful will be written as $[wB,wB]\le H^p,\forall w\in W.$ □

Corollary 2.

As with the notation above, the base group of wreath products is a powerful p-group. In addition, $\mathrm{\Theta }\left(B\right)$ is powerfully embedded in B.

Proof. 

The definition of the base group of wreath product, according to Lemma 9(i), is a direct product of powerful groups. Hence, it is a powerful p-group. As Corollary 1, $\mathrm{\Theta }\left(B\right)$ is powerfully embedded in B. □

Corollary 3.

The center of the base group of a wreath product is a powerful p-group.

Proof. 

Let $Z\left(B\right)$ be the center of the base group. As in Remark 2, it is a cyclic group, which is abelian. In particular, it is powerful. □

Example 1.

Consider wreath product $W\cong C_3\wr C_3$ such that B is extra-special 3-group of order 27, exponent 9. Then B is a powerful 3-subgroup. Since $B^3=[B,B]$. Let H be a 3-subgroup of order 3, thus H is cyclic, then H is an abelian group. Hence, H is powerful.

The following lemma shows that the commutator subgroup of a wreath product is powerful.

Lemma 10.

Let B be a powerfully embedded subgroup of $W=B⋊H$. Then, the commutator subgroup of wreath product W is powerful.

Proof. 

As B is a normal subgroup of the wreath product W. Then, $[B,W]$ is a subgroup of B and $[B,W]$ is a normal subgroup of W. Since B is a powerfully embedded subgroup of W. Then, $[B,W]$ is powerfully embedded in W. Let $[W,W]=W^{\prime }$ and $[B,W]=N$. Now, we have that $N\le W^{\prime }\le W$. Since $N\le B$, we find the order of N is equal to $p^{p-1}$ and of $W^{\prime }/N$ is equal to p or e. Which means it is cyclic group. Hence, $W^{\prime p}\ge N^p\ge [N,W^{\prime }]=[W^{\prime },W^{\prime }]$. Hence, $W^{\prime }=[W,W]$ is powerful. □

Example 2.

The wreath product $W=C_3\wr C_3$ is a metablian group, then the commutator subgroup is abelian. Hence, it is powerful.

Lemma 11.

Let B be a powerfully embedded subgroup of $W=G\wr H$. Let A be a subgroup of H. If $B\le G\wr A\le W$ and $(G\wr A)/B$ is cyclic, then $G\wr A$ is powerful.

Proof. 

Since ${\displaystyle \frac{|G\wr A|}{\left|B\right|}}={\displaystyle \frac{\left|B\right|\left|A\right|}{\left|B\right|}}=\left|A\right|$, thus $(G\wr A)/B\cong A$. A subgroup of cyclic is cyclic. Then, $G\wr A/B$ is cyclic. Hence, ${(G\wr A)}^p\ge B^p\ge [B,G\wr A]=[G\wr A,G\wr A],$ so $G\wr A$ is powerful. □

3.2. Powerful Wreath Product

In this section, we are concerned with creating the necessary and sufficient conditions for a wreath product to be powerful and quasi-powerful p-group. Let p be a prime number. Suppose that the base group $B=\underset{indexbyp}{\overset{G\times G\times \cdots \times G}{︸}}$ is powerfully embedded in the wreath product.

Remark 5.

(1) 

An elementary abelian group is defined as an abelian group that all elements, except for the identity element, possess at the same order.

(2) 

Recall that every finite p-group is nilpotent, and p is a prime number. The wreath product has order $p^n$, where $n=p+1$, so the wreath product is a p-group, which means nilpotent.

Definition 14.

Assume p is a prime number. The wreath product of two permutation cyclic p-groups is called a quasi-powerful p-group.

The following lemma shows that the above definition exists.

Lemma 12.

Consider wreath product $W=C_p\wr C_p$. Then, W is a quasi-powerful p-group.

Proof. 

Let p be aprime number. Let $W=C_p\wr C_p$. Then, $Z\left(W\right)=C_p$ and $W/Z\left(W\right)=C_p\times C_p$, which is a p-group and elementary abelian. Then, it is powerful. Hence, W is a quasi-powerful p-group. □

Example 3.

For $C_2\wr C_2$, we have that $W/Z\left(W\right)=C_2\times C_2$. Then, the commutator subgroup $[C_2\times C_2,C_2\times C_2]=[C_2,C_2]\times [C_2,C_2]=e\times e=e_{W/Z\left(W\right)}.$ Hence, $[W/Z\left(W\right),W/Z\left(W\right)]\le {(W/Z\left(W\right))}^4.$ Finally, $W/Z\left(W\right)$ is powerful and $C_2\wr C_2$ is a quasi-powerful p-group.

Corollary 4.

Consider wreath product $W=C_p\wr C_p$, if $p>3,$ then W is potent.

Proof. 

For any prime number p greater than 3, it is a known fact that every p-group that is quasi-powerful is also potent. By applying Lemma 12, it follows that W is potent. □

Definition 15.

Let $p>3$ and p be a prime number. The wreath product of cyclic p-groups with base group B is powerfully embedded as a powerful wreath product p-group.

In the following lemma, we show that under certain conditions, the wreath product must be powerful.

Lemma 13.

Let $p>3$ and p be a prime number. If B is a powerfully embedded subgroup of $W=B⋊H$, then the wreath product W is a powerful p-group.

Proof. 

Suppose that $p>3$ and p is a prime number. As in Lemma 5, since B is powerfully embedded. Then, $[K,K]=[\langle w\rangle B,\langle w\rangle B]\le B^p,\forall w\in W.$ As in Lemma 9(ii), we obtain $[wB,wB]\le H^p,\forall w\in W$. Now,

$$\begin{array}{cc}\hfill & [wB,wB]·[\langle w\rangle B,\langle w\rangle B]\le H^p·B^p,\hfill \\ \hfill & \Rightarrow [w\langle w\rangle ,w\langle w\rangle ]B\le W^p\hfill \\ \hfill & \Rightarrow [w,w]B\le W^p\hfill \\ \hfill & \Rightarrow B\le W^p.\hfill \end{array}$$

(5)

Since $W/B$ is an abelian, thus $[W,W]\le B.$ Therefore, $[W,W]\le B\le W^p,$ which means $W/W^p$ is an abelian. Hence, W is powerful p-group. □

The following examples give us the reason to choose p as an odd prime and $p>3$.

Example 4.

When $p=2,$ there is certainly no powerful wreath product defined. There are many instances of 2-groups for which all groups constructed wreath products and are powerful (even abelian), but the wreath product itself is not, for example the dihedral group of order $8.$

Example 5.

When $p=3,$ there is certainly no powerful wreath product defined. Let us consider the permutation groups $C=\left\{\right(1),(135),(153\left)\right\}$ and $D=\left\{\right(1),(246),(264\left)\right\}$ acting on the sets $X=\{1,3,5\}$ and $Y=\{2,4,6\}$, respectively. Let $W=C\wr D$. Details are in Reference [7]. Then, $|W^3|={\displaystyle \frac{\left|W\right|}{|{\mathrm{\Omega }}_3|}}=1$. The commutator subgroup is not contained in $W^3$.

Lemma 14.

Let $(G,X)$ and $(H,Y)$ be two permutation groups and A be a normal subgroup of G. Then,

$$(G\wr H)/A^Y\cong (G/A)\wr H.$$

Proof. 

See Theorem 4.13 in [24]. □

Lemma 15.

Let $(G,X)$ and $(H,Y)$ be two permutation cyclic p-groups and A be a normal subgroup of $G.$ If A is powerful and B is a powerfully embedded subgroup of W, then $(G/A)\wr H$ is powerful.

Proof. 

As Lemma 14, $(G\wr H)/A^Y\cong (G/A)\wr H.$ Since B is a B powerfully embedded subgroup of W, then $G\wr H$ is powerful. Since $A^Y$ is a direct product of powerful, then it is powerful. It is obvious that factor groups of powerful are powerful. Hence, $(G/A)\wr H$ is powerful. □

Lemma 16.

Let G and H be finite cyclic p-groups that act on finite sets X and Y, respectively, such that there exists a homomorphism function from G into H. Let $W=G\wr H$ be a group act on $X\times Y$. Then,

$$W/(ker\theta \wr H)\cong (Im\theta \wr H).$$

Proof. 

Assume that $\theta :G⟶H$ is a homomorphism map. By First Isomorphism Theorom of groups, we have that

$$G/ker\theta \cong Im\theta$$

Since $ker\theta$ is a subgroup of G, then $ker\theta \wr H$ can be embedded in a natural way in $G\wr H$ by Lemma 8. Thus, we can defined:

$$F:ker\theta \wr H\to W$$

as an injective, given by $F\left(k\right)=w,$ such that $k\in ker\theta \wr H,$ and $w\in W,$ with $k=sh,w=gh,$ where $s\in ker\theta ,g\in G,$ and $h\in H$ and is a homomorphism. Since

$$\begin{array}{cc}\hfill F\left(k{k}^{\prime }\right)& =F\left(sh{s}^{\prime }{h}^{\prime }\right)\hfill \\ \hfill & =F\left(s{s}^{\prime -h}h{h}^{\prime }\right)\hfill \\ \hfill & =g{g}^{\prime -h}h{h}^{\prime }\hfill \\ \hfill & =w{w}^{\prime }\hfill \\ \hfill & =F\left(k\right)F\left(k^{\prime }\right),\hfill \end{array}$$

where $k,k^{\prime }\in ker\theta \wr H,$ and $w,w^{\prime }\in W.$ Then, $ker\theta \wr H\cong ImF⩽W.$ This injective homomorphism let us think of $ker\theta \wr H$ as a subgroup of W. Now, we show that $ker\theta \wr H$ is normal in $W,$ follow from the calculation,

$$\begin{array}{cc}\hfill e_{G}hkh{\left(e_{G}h\right)}^{-1}& =e_{G}hkh{e}_G^{h^{-1}}{h}^{-1}\hfill \\ \hfill & =e_{G}hkh{e}_G{h}^{-1}\hfill \\ \hfill & =e_{G}hk{e}_G^{h^{-1}}h{h}^{-1}\hfill \\ \hfill & =e_{G}hk{e}_H\hfill \\ \hfill & =e_G{k}^{h^{-1}}h{e}_H\hfill \\ \hfill & =k^{h^{-1}}h\in ker\theta \wr H.\hfill \end{array}$$

Since $ker\theta \wr H⊴W$, then $W/ker\theta \wr H$ is a quotient group of W of order $|G:ker\theta |.$ Now, since $ker\theta ⊴G,$ then by Lemma 14 and The First Isomorphism Theorem, we find that

$$W/{\left(ker\theta \right)}^Y\cong (G/ker\theta )\wr H\cong Im\theta \wr H,$$

let $\psi :W\to Im\theta \wr H,$ with $ker\psi ={\left(ker\theta \right)}^Y.$ Thus,

$$\begin{array}{cc}\hfill {\left(ker\theta \right)}^Y& =ker\psi \hfill \\ \hfill & =\{gh\in G\wr H:\psi (gh)=e\}\hfill \\ \hfill & =\{gh:\widehat{g}h=e,\widehat{g}\in Im\theta \}\hfill \\ \hfill & =\{gh:\widehat{g}\left(y\right)h=e,\forall y\in Y\}\hfill \\ \hfill & =\{gh:g(y)\in ker\theta ,\forall y\in Y\},\hfill \end{array}$$

from the definition of $\widehat{g}.$ Hence, ${\left(ker\theta \right)}^Y=ker\theta \wr H.$ Therefore,

$$W/ker\theta \wr H\cong Im\theta \wr H.$$

□

Lemma 17.

Let G and H be finite cyclic p-groups that act on finite sets X and Y, respectively, such that there exists homomorphism function from G into H. Let $W=G\wr H$ be a group act on $X\times Y$. If $ker\theta$ is powerful and B is powerfully embedded in $W,$ then $(W/ker\theta )\wr H$ is powerful.

Proof. 

As Lemma 20, we have that $(G/ker\theta )\wr H$ is powerful. As Lemma 16, clearly we have $W/(ker\theta \wr H)$ is powerful. □

Lemma 18.

Let H be a ${\mu }_i$ group of exponent p, where p is odd prime number and $i\in \mathbb{N}.$ If B is a powerfully embedded subgroup of $W=G\wr H\cong B⋊H$. Then, W is ${\mu }_i$ group of exponent p.

Proof. 

Assume that p is an odd prime number. Since H be a ${\mu }_i$ group of exponent p, $i\in \mathbb{N}.$ Then, all subgroups of index $p^i$ are powerful of exponent p. without loss generalization, suppose that A is a subgroup of H such that it is powerful. So, $[H,A]=p^i$ and $[H,H]\le H^p$. As $A\le H$. Then, $G\wr A\le G\wr H$ and $|G\wr H:G\wr A|={\displaystyle \frac{\left|H\right|}{\left|A\right|}}=p^i$. By Lemma 11, we have that $G\wr A$ is powerful. Therefore, W is a ${\mu }_i$ group of exponent p. □

The following lemma gives us the correlation between a subgroup of wreath product, a subgroup of base wreath product, and a subgroup of a group-constructed wreath product such that all subgroups are generated by all $p^{i}th$ powers $\forall i\in \mathbb{N},$ as in Remark 3.

Lemma 19.

If B and H are powerfully embedded in $W=G\wr H=B⋊H,$ then

$$W^{p^i}=B^{p^i}{H}^{p^i},i\in \mathbb{N}$$

(6)

Proof. 

Suppose that p is an odd prime. Let $i=1,B^p{H}^p\le {\left(BH\right)}^p=W^p.$ In $W/B^p{H}^p$ both B and H become central. They have exponent p. So, $W^p={\left(BH\right)}^p\le B^p{H}^p.$ The general case is obtained by simple induction. □

Corollary 5.

If B and H are powerfully embedded in $W=G\wr H=B⋊H.$ Then, $[W^{p^i},W]\le B^{p^{i+1}}{H}^{p^{i+1}}\forall i\in \mathbb{N}.$

Proof. 

Observably, from Lemmas 6 and 19, we obtain the required. □

The following lemma relates the Frattini subgroup of the base group that forms the wreath product to the Frattini subgroup of the wreath product.

Lemma 20.

If B and H are powerfully embedded subgroups in $W=G\wr H=B⋊H$. The Frattini subgroup of base group constitute the wreath product is isomorphic to the Frattini subgroup of the wreath product.

Proof. 

Suppose that $p>3$ and p is a prime number. Since B and H are powerfully embedded subgroups in $W=G\wr H$, thus they are also powerful p-groups. Then, for $i\in \mathbb{N},\mathrm{\Theta }\left(B\right)=B^{p^i}.$ As in Lemma 19, we have $W^{p^i}=B^{p^i}{H}^{p^i},i\in \mathbb{N}$. As in Theorem 13, the wreath product W is powerful. Then, for $i\in \mathbb{N},\mathrm{\Theta }\left(W\right)=W^{p^i}$. Since H is a cyclic group of order p. Then, $H^p=\left\{e\right\}.$ Now,

$$\begin{array}{cc}\hfill \mathrm{\Theta }\left(W\right)& =W^{p^i}\hfill \\ \hfill & =B^{p^i}{H}^{p^i}\hfill \\ \hfill & =B^{p^i}e\hfill \\ \hfill & =B^{p^i}\hfill \\ \hfill & =\mathrm{\Theta }\left(B\right).\hfill \end{array}$$

(7)

□

Lemma 21.

Let $W=G\wr H$ be the wreath product as a p-group. Then, the commutator subgroup of the wreath product is a subgroup of $\mathrm{\Theta }\left(B\right)$ such that $p>3$ and p is a prime number.

Proof. 

As Remark 5(2), W is a finite nilpotent and B a maximal normal subgroup of W. Since $W/B$ is an isomorphic to cyclic group $H,$ in a particular abelian. Then, $[W,W]\le B$. Then, $[W,W]\le \mathrm{\Theta }\left(W\right).$ By Lemma 20, $\mathrm{\Theta }\left(W\right)=\mathrm{\Theta }\left(B\right)$. Hence, $[W,W]\le \mathrm{\Theta }\left(B\right).$ □

4. Discussion

This section provides a comprehensive analysis of our research, with a specific emphasis on a noteworthy interpretation concerning the relationship between the wreath product and a p-group with significant power. The present study offers an academic analysis of core principles in the domain of group theory, with a particular emphasis on subjects including the wreath product and powerful p-groups. In this study, we investigate the structural properties of the wreath product of cyclic p-groups, with particular emphasis on the groups that are powerfully embedded within this product. The main finding of this study concerns the development of the powerful wreath product and the quasi-powerful wreath product. This study aims to demonstrate the significant influence of subgroups that are powerful within the wreath product, with a specific emphasis on p-groups. The aforementioned result is based on the assumption that p is a prime number and W is the standard wreath product of two nontrivial cyclic p-groups, denoted as G and H. This study exhibits potential for future expansion by incorporating supplementary research and investigations in subsequent endeavors. In the field of group theory, there are numerous unresolved questions and difficulties surrounding wreath products involving p-groups with powerful properties. There are multiple unresolved inquiries and calculations in this domain that may necessitate further investigation.

5. Conclusions

The present study offers an academic analysis of core principles in the domain of group theory, with a particular emphasis on subjects like the wreath product and powerful p-groups. In this study, we analyze the structural properties of the wreath product formed by cyclic p-groups, paying particular attention to the groups that are powerfully embedded within this product. The main finding of this study concerns the development of the powerful wreath product and the quasi-powerful wreath product. In this work, we provide evidence that supports the claim that specific subgroups of the wreath product possess the defining characteristics of a powerful p-group. The existence of instances of the influential subgroup within the wreath product can be observed. Lemma 9 demonstrates that groups that construct the wreath product exhibit the characteristic of being powerful p-groups. The center of the base group of a wreath product is a powerful p-group. The Lemma 10 demonstrates that the commutator subgroup of a wreath product possesses the property of being powerful. One of the motivations behind this work is to introduce new classes of potent p-groups known as powerful wreath products and quasi-powerful wreath products. Our primary focus revolves around the establishment of the essential and comprehensive criteria that determine whether a wreath product possesses the attributes of being powerful and quasi-powerful. The aforementioned result is based on the assumption that p is a prime number and W is the standard wreath product of two nontrivial cyclic p-groups, denoted as G and H. Lemma 12 says the wreath product of two permutation cyclic p-groups is called a quasi-powerful p-group. Also, Lemma 13 shows that under certain conditions, the wreath product must be powerful and Lemma 18 shows W is ${\mu }_i$ group of exponent p. In addition, Lemma 17 assume that G and H be finite cyclic p-groups act on finite sets X and Y, respectively, such that there exists a homomorphism function from G into H. Let $W=G\wr H$ be a group act on $X\times Y$. If $ker\theta$ is powerful and B is powerfully embedded in $W,$ then $(W/ker\theta )\wr H$ is powerful. Finally, if the B and p-subgroup H are powerfully embedded, we can draw two conclusions. First, it gives us the correlation between a subgroup of wreath products, a subgroup of base wreath products, and a subgroup of group-constructed wreath products such that all subgroups are generated by all $p^i$-th powers, $\forall i\in \mathbb{N}$. The second result relates the Frattini subgroup of the base group that forms the wreath product to the Frattini subgroup of the wreath product. Our future investigation is going to focus on the Generalized Wreath Product of Powerful p-groups. We refer to the use of the citations [25,26].