The Normalizer Property for Integral Group Rings of Generalized Twisted Wreath Products of Two Finite Groups

Abstract

Let N and H be two finite groups. In this paper, we define a generalized twisted wreath product $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ and establish sufficient conditions for G to satisfy the normalizer property. Additionally, we investigate the outer class-preserving and Coleman automorphism groups of certain generalized twisted wreath products. Our results extend several known theorems in the literature.

1. Introduction

All groups considered in this paper are finite. Let G be a group and $\mathbb{Z}G$ its integral group ring over $\mathbb{Z}$. A long-standing problem [1] in the theory of integral group rings, known as the normalizer problem, asks whether

$$N_{U\left(\mathbb{Z}G\right)}\left(G\right)=G·C_{U\left(\mathbb{Z}G\right)}\left(G\right)$$

holds for every group G, where $N_{U\left(\mathbb{Z}G\right)}\left(G\right)$ and $C_{U\left(\mathbb{Z}G\right)}\left(G\right)$ denote the normalizer and centralizer of G in the unit group $U\left(\mathbb{Z}G\right)$, respectively. If this equality is satisfied, we say that G has the normalizer property.

This problem can be reformulated in terms of automorphisms. For any unit $u\in N_{U\left(\mathbb{Z}G\right)}\left(G\right)$, the map ${\sigma }_u{|}_G:g\mapsto u^{-1}gu$ defines an automorphism of G. The set of all such automorphisms forms a subgroup ${Aut}_{\mathbb{Z}}\left(G\right)$ of $Aut\left(G\right)$. As shown by Jackowski and Marciniak [2], the normalizer property is equivalent to the condition ${Aut}_{\mathbb{Z}}\left(G\right)=Inn\left(G\right)$; equivalently, ${Out}_{\mathbb{Z}}\left(G\right)={Aut}_{\mathbb{Z}}\left(G\right)/Inn\left(G\right)$ is trivial.

Two important subgroups of $Aut\left(G\right)$ play a central role in this field: the group ${Aut}_{\mathrm{c}}\left(G\right)$ of class-preserving automorphisms (those preserving every conjugacy class of G), and the group ${Aut}_{\mathrm{Col}}\left(G\right)$ of Coleman automorphisms (those whose restriction to any Sylow subgroup coincides with the restriction of some inner automorphism). Let ${Out}_{\mathrm{c}}\left(G\right)={Aut}_{\mathrm{c}}\left(G\right)/Inn\left(G\right)$ and ${Out}_{\mathrm{Col}}\left(G\right)={Aut}_{\mathrm{Col}}\left(G\right)/Inn\left(G\right).$ It is known that ${Out}_{\mathbb{Z}}\left(G\right)\le {Out}_{\mathrm{c}}\left(G\right)\cap {Out}_{\mathrm{Col}}\left(G\right)$ and that ${Out}_{\mathbb{Z}}\left(G\right)$ is an elementary abelian 2-group [2,3]. Therefore, if ${Out}_{\mathrm{c}}\left(G\right)\cap {Out}_{\mathrm{Col}}\left(G\right)$ has odd order, then G has the normalizer property. Although Hertweck [4] constructed a counterexample to the normalizer problem in general, it remains an active and fruitful endeavor to identify families of groups for which the property holds.

In the present work, we continue this line of investigation by studying the normalizer property for a broader class of groups constructed via generalized twisted wreath products. Such constructions naturally arise in the study of group extensions and possess rich automorphism-theoretic properties. Our aim is to establish sufficient conditions under which these groups satisfy the normalizer property, building on and extending the techniques developed in earlier work on wreath products. The reader may refer to [5,6,7,8].

The paper is organized as follows. In Section 2, we give the definition of generalized twisted wreath products of two groups and recall preliminary results on the normalizer problem, as well as class-preserving and Coleman automorphisms. In Section 3, we study the normalizer property of generalized twisted wreath products and some other extensions. In Section 4, we focus on the class-preserving Coleman automorphisms of generalized twisted wreath products. Our results generalize and unify some earlier work on wreath products.

The notation and terminology in this paper are standard and agree with the book [9]. Below we list some frequently used notation. We denote by ${\sigma }_g$ the inner automorphism of a group G induced by $g\in G$, namely, $x^{{\sigma }_g}=g^{-1}xg$ for all $x\in G$. For a fixed prime p, $O_p\left(G\right)$ denotes the largest normal p-subgroup of G, and $Z\left(G\right)$ denotes the center of G, both of which are characteristic in G.

2. Preliminaries

We start by introducing the generalized twisted wreath product. Actually, the standard wreath product, the regular wreath product, and the twisted wreath product in group theory can all be viewed as special cases of this generalized construction. Consequently, all results established in this paper apply to these three particular situations.

Definition 1.

Let N and H be groups, and fix a positive integer k. Denote by $\mathcal{S}=\{U_1,\dots ,U_k\}$ a set of subgroups of H, and by $\mathcal{T}=\{{\tau }_1,\dots ,{\tau }_k\}$ a set of mutually independent homomorphisms, where each ${\tau }_i:U_i\to Aut\left(N\right)$. For each $i\in \{1,2,\cdots ,k\}$, choose a transversal $R_i$ of $U_i$ in H, and let $R=R_1\bigsqcup \cdots \bigsqcup R_k$ be their disjoint union. We define the generalized twisted wreath product $N{\wr }_{\mathcal{S},\mathcal{T},R}H$ of N with H, which depends on the triple $(\mathcal{S},\mathcal{T},R)$, to be the set

$$\left\{\right(f,h)\mid h\in H,f:R\to N\}$$

with the following multiplication:

For each $r\in R$, there exists a unique index i such that $r\in R_i$. If $h_1,h_2\in H$ and $r{h}_1=u^{\prime }{r}^{\prime }$ with $u^{\prime }\in U_i$ and $r^{\prime }\in R_i$, then

$$(f_1,h_1)(f_2,h_2)=(f^{\prime },h_1{h}_2),$$

where $f^{\prime }\left(r\right)=f_1\left(r\right)f_2{\left(r^{\prime }\right)}^{{\tau }_i\left(u^{\prime -1}\right)}$.

It can be verified by a direct calculation that this multiplication is associative. The identity element is $(e,1)$, where $e\left(r\right)=1$ for all $r\in R$. In addition, the inverse of $(f,h)$ is $(\overline{f},h^{-1})$, where

$$\overline{f}\left(r^{\prime }\right)={\left(f{\left(r\right)}^{-1}\right)}^{{\tau }_i\left(u^{\prime }\right)}$$

for $rh=u^{\prime }{r}^{\prime }$, with $r,r^{\prime }\in R_i$, $u^{\prime }\in U_i$, and $i\in \{1,2,\cdots ,k\}$. Therefore, $N{\wr }_{\mathcal{S},\mathcal{T},R}H$ is a group.

Remark 1.

Let $G=N{\wr }_{\mathcal{S},\mathcal{T},R}H$ be a generalized twisted wreath product in Definition 1. Then one can verify the following statements.

(1) 

If $k=1$, then the generalized twisted wreath product coincides with the twisted wreath product in Definition 15.10 in Huppert’s book [9].

(2) 

Let $R=R_1\bigsqcup \cdots \bigsqcup R_k$ and $R^{\prime }=R_1^{\prime }\bigsqcup \cdots \bigsqcup R_k^{\prime }$ be two disjoint unions, where $R_i$ and $R_i^{\prime }$ are two transversals of $U_i$ in H for each i. A straightforward calculation shows that $N{\wr }_{\mathcal{S},\mathcal{T},R}H\cong N{\wr }_{\mathcal{S},\mathcal{T},R^{\prime }}H$, where the isomorphic map is defined by $(f,h)\mapsto (\tilde{f},h)$, with $\tilde{f}\left(\tilde{r}\right)=f{\left(r\right)}^{{\tau }_i\left(u_r^{-1}\right)}$ if $r\in R_i,\tilde{r}\in R_i^{\prime }$, and $\tilde{r}=u_{r}r$ for some $u_r\in U_i$, with $i\in \{1,2,\cdots ,k\}$. Therefore, up to isomorphism, the generalized twisted wreath product in Definition 1 depends only on $\mathcal{S}$ and $\mathcal{T}$. Consequently, we can denote it simply by $N{\wr }_{\mathcal{S},\mathcal{T}}H$.

(3) 

The conjugation action of $\left\{\right(e,h)\mid h\in H\}$ on $\left\{\right(f,1)\mid f:R\to N\}$ is

$${(f,1)}^{(e,h)}=(f^{\prime },1),$$

where $f^{\prime }\left(r\right)=f{\left(r^{\prime }\right)}^{{\tau }_i\left(u^{\prime -1}\right)}$ if $r\in R_i$ and $r{h}^{-1}=u^{\prime }{r}^{\prime }$ with $u^{\prime }\in U_i$ and $r^{\prime }\in R_i$. For convenience, we identify $(f,1)$ and $(e,h)$ with f and h, respectively. Write $B=\{f\mid f:R\to N\}$. Then we see that $N{\wr }_{\mathcal{S},\mathcal{T}}H=B⋊H$, and B is called the base group of the generalized twisted wreath product.

(4) 

Let $N_r=\{f\mid f\left(r\right)\in Nandf\left(s\right)=1if s\ne r\}$. Clearly, B is the direct product of $N_r$ for all $r\in R$, namely, $B={\times }_{r\in R}{N}_r$. To be more specific, if $b\in B$ satisfies $b\left(r\right)\in N$ for every $r\in R$, then $b={\prod }_{r\in R}{b}_r$, where each $b_r\in N_r$ satisfies $b_r\left(r\right)=b\left(r\right)$.

Lemma 1.

Let $G=N{\wr }_{\mathcal{S},\mathcal{T}}H=B⋊H$ be the generalized twisted wreath product of N with H as in Definition 1. Let $N_r$ be the r-th direct factor of the base group B of G as in Remark 1. Suppose that ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$. If $V\le B$ such that $V\cap N_r>1$ for all $r\in R$, then $C_G\left(V\right)\le B$.

Proof. 

Let $bh\in C_G\left(V\right)$ with $b\in B$ and $h\in H$. Suppose that $h\ne 1$. Then there exists some $i_0$ such that $h\notin {\mathrm{Core}}_H\left(U_{i_0}\right)$. Write $R=R_1\bigsqcup R_2\bigsqcup \cdots \bigsqcup R_k$. Hence, there exist distinct $r,r^{\prime }\in R_{i_0}$ such that $r{h}^{-1}=u^{\prime }{r}^{\prime }$ with $u^{\prime }\in U_{i_0}$. Choose $f\in V\cap N_r$ such that $f\left(r\right)\ne 1$. If $f^b=f^{\prime }$, then $f^{\prime }\left(r\right)=f{\left(r\right)}^{b\left(r\right)}\ne 1$ and $f^{\prime }\left(s\right)=1^{b\left(s\right)}=1$ for $s\ne r$. Therefore, we obtain

$$f=f^{bh}=f^{\prime h}=f^{\prime \prime }.$$

By Remark 1 (3), $f^{\prime \prime }\left(r\right)=f^{\prime }{\left(r^{\prime }\right)}^{{\tau }_{i_0}\left(u^{\prime -1}\right)}=1\ne f\left(r\right)$, which contradicts $f=f^{\prime \prime }$. □

Lemma 2

([1] Proposition 1.4). Let G be a group. Suppose $\gamma =\mathsf{\Sigma }\gamma \left(g\right)g\in \mathbb{Z}G$ is a torsion unit; namely, ${\gamma }^n=1$ for some n and $\gamma \left(1\right)\ne 0$. Then $\gamma =\pm 1$.

Lemma 3

([1] Proposition 9.5). Let $u\in N_{\mathrm{U}\left(\mathbb{Z}G\right)}\left(G\right)$ and let ${\sigma }_u$ be the automorphism of G induced by u via conjugation. Then ${\sigma }_u^2\in Inn\left(G\right)$.

Lemma 4

([5] Proposition 3.1). Let P be a p-subgroup of a group G, $N⊴G$, and let $u\in N_{\mathrm{U}\left(\mathbb{Z}G\right)}\left(G\right)$. Suppose that $\rho \left(u\right)=N{g}_1\in G/N$ for some $g_1\in G$, where $\rho :\mathbb{Z}G\to \mathbb{Z}(G/N)$ is the natural ring homomorphism. Then there exists $n_0\in N$ such that $u^{-1}xu={\left(n_0{g}_1\right)}^{-1}x\left(n_0{g}_1\right)$ for all $x\in P$.

Lemma 5

([10] Lemma 2). Let p be a prime, G a group, and φ an automorphism of G of p-power order. Suppose that there is a normal subgroup N of G such that φ fixes all elements of N, and that φ induces the identity on the quotient group $G/N$. Then φ induces the identity on $G/O_p(Z\left(N\right))$. Further, if φ fixes element-wise a Sylow p-subgroup of G, then φ is an inner automorphism of G.

Lemma 6

([11] Lemma 6). Let N be a normal subgroup of a group G and let $\phi \in Aut\left(G\right)$ such that $N^{\phi }=N$. Let Q be a Sylow subgroup of N. Suppose that ${\phi |}_Q={\sigma }_h{|}_Q$ for some $h\in G$. Then φ fixes $M:=N{C}_G\left(Q\right)$ and ${\phi |}_{G/M}={\sigma }_h{|}_{G/M}$.

Lemma 7

([8] Lemma 2.7). Let G be a group with $M\le G$ and $N⊴G$. Let p be a prime and let $\varphi \in Aut\left(G\right)$ be of p-power order with $N^{\varphi }=N$. Then the following statements hold.

(1) 

Assume that ${\varphi |}_M={\sigma }_g{|}_M$ for some $g\in G$. Then there exists a $\phi \in Aut\left(G\right)$ of p-power order such that ${\phi |}_M=id$. In fact, $\phi Inn\left(G\right)$ is a power of $\varphi Inn\left(G\right)$ in $Out\left(G\right)$. Furthermore, $\varphi \in Inn\left(G\right)$ if and only if $\phi \in Inn\left(G\right)$;

(2) 

Assume that ${\varphi |}_{G/N}={\sigma }_g{|}_{G/N}$ for some $g\in G$. Then there exists a $\psi \in Aut\left(G\right)$ of p-power order such that ${\psi |}_{G/N}=id$. In fact, $\psi Inn\left(G\right)$ is a power of $\varphi Inn\left(G\right)$ in $Out\left(G\right)$. Furthermore, $\varphi \in Inn\left(G\right)$ if and only if $\psi \in Inn\left(G\right)$.

Lemma 8

([8] Lemma 2.8). Let G be a group with $M\le G$ and $N⊴G$. Let p be a prime and let $\varphi \in Aut\left(G\right)$ be of p-power order with $M^{\varphi }=M$ and $N^{\varphi }=N$. Then the following statements hold.

(1) 

If ${\varphi |}_M={\sigma }_{g_1}{|}_M$ for some $g_1\in G$, then there exists some p-element $x_1\in G$ such that ${\varphi |}_M={\sigma }_{x_1}{|}_M$;

(2) 

If ${\varphi |}_{G/N}={\sigma }_{g_2}{|}_{G/N}$ for some $g_2\in G$, then there exists some p-element $x_2\in G$ such that ${\varphi |}_{G/N}={\sigma }_{x_2}{|}_{G/N}$.

Remark 2.

Let G be a group and $\phi \in Aut\left(G\right)$ of order $p^{m}n$, with p a prime and $(p,n)=1$. Suppose that ${\phi }^{p^k}\in Inn\left(G\right)$ for some k. Then $\phi \in Inn\left(G\right)$ if and only if ${\phi }^n\in Inn\left(G\right)$. In particular, if we want to show $\phi \in Inn\left(G\right)$, then it suffices to assume that φ is of p-power order.

3. The Normalizer Property of Generalized Twisted Wreath Products

Inspired by the work of Hai and Guo in [12], we obtain the following foundational proposition.

Proposition 1.

Let G be a group with a normal subgroup N. Suppose that all central units of $\mathbb{Z}(G/N)$ are trivial. Let $\phi \in {Aut}_{\mathbb{Z}}\left(G\right)$. Then there exists an $x\in G$ such that ${\varphi |}_N\in {Aut}_{\mathrm{Col}}\left(N\right)$, where $\varphi =\phi {\sigma }_x$. Furthermore, let P be a Sylow p-subgroup of G. Then there exists a $y\in G$ such that $\psi =\phi {\sigma }_y$ satisfies ${\psi |}_{G/N}{=id|}_{G/N}$, ${\psi |}_P{=id|}_P$ and ${\psi |}_N\in {Aut}_{\mathrm{Col}}\left(N\right)$.

Proof. 

Let $\phi \in {Aut}_{\mathbb{Z}}\left(G\right)$. Then there exists some $u\in N_{U\left(\mathbb{Z}G\right)}\left(G\right)$ such that $\phi ={\sigma }_u{|}_G$. Without loss of generality, we may assume that u is of augmentation 1. Since $\phi$ is a class-preserving automorphism, we see that $\phi$ induces an automorphism of $G/N$, denoted by ${\phi |}_{G/N}$. Let $\rho :\mathbb{Z}G\to \mathbb{Z}(G/N)$ be the natural ring homomorphism. Thus, $\rho \left(u\right)\in N_{U\left(\mathbb{Z}\overline{G}\right)}\left(\overline{G}\right)$. It follows that

$${\phi |}_{G/N}={\sigma }_{\rho \left(u\right)}{|}_{G/N}.$$

(1)

By Lemma 3 and (1), ${\sigma }_{\rho {\left(u\right)}^2}{=\sigma |}_{\overline{h_0}}$ for some $h_0\in G$. Since all central units of $\mathbb{Z}(G/N)$ are trivial, we see that $\rho {\left(u\right)}^2{\overline{h_0}}^{-1}\in G/N$, namely, $\rho {\left(u\right)}^2\in G/N$.

Set $\overline{g_0}\in supp\left(\rho \left(u\right)\right)$. Then $\rho \left(u\right){\overline{g_0}}^{-1}G/N=\rho \left(u\right)G/N$ is of order 2. This implies ${\left(\rho \left(u\right){\overline{g_0}}^{-1}\right)}^2\in G/N$, and therefore, $\rho \left(u\right){\overline{g_0}}^{-1}$ has finite order by the finiteness of $G/N$. Furthermore, note that the coefficient of $\overline{1}$ in $\rho \left(u\right){\overline{g_0}}^{-1}$ is nonzero, and the augmentation of $\rho \left(u\right)$ is 1. Lemma 2 implies

$$\rho \left(u\right)=\overline{g_0}\in G/N.$$

(2)

Set $x=g_0^{-1}$. Let $N_p$ be a Sylow p-subgroup of N with $p\in \pi \left(N\right)$. According to Lemma 4, there exists some $n\in N$ such that ${\varphi |}_{N_p}={\sigma }_n{|}_{N_p}$. Since $p\in \pi \left(N\right)$ is arbitrary, it follows that $\varphi \in {Aut}_{\mathrm{Col}}\left(N\right)$.

By (1) and (2), we obtain

$${\varphi |}_{G/N}{=id|}_{G/N}.$$

(3)

Notice that P is a Sylow p-subgroup of G. By Lemma 4, there exists some $m\in N$ such that

$${\varphi |}_P={\sigma }_m{|}_P.$$

(4)

Set $y=g_0^{-1}{m}^{-1}$ and $\psi =\phi {\sigma }_y$. According to (3) and (4), the proof is complete. □

Proposition 1 takes the following more refined result when applied to generalized twisted wreath products.

Proposition 2.

Let $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ be the generalized twisted wreath product of N, with H as in Definition 1. Suppose that all central units of $\mathbb{Z}H$ are trivial, and ${\tau }_i\left(U_i\right)\le {Aut}_{\mathrm{c}}\left(N\right)$ for each i. Let $\phi \in {Aut}_{\mathbb{Z}}\left(G\right)$ and $N_r$ be the r-th direct factor of the base group B of G, as in Remark 1. Then there exists some $x\in G$ such that $\left(\phi {\sigma }_x\right){|}_{N_r}\in {Aut}_{\mathrm{c}}\left(N_r\right)\cap {Aut}_{\mathrm{Col}}\left(N_r\right)$. Furthermore, let P be a Sylow p-subgroup of G. Then there exists a $y\in G$ such that $\left(\phi {\sigma }_y\right){{|}_{G/B}=id|}_{G/B}$, $\left(\phi {\sigma }_y\right){{|}_P=id|}_P$ and $\left(\phi {\sigma }_y\right){|}_{N_r}\in {Aut}_{\mathrm{c}}\left(N_r\right)\cap {Aut}_{\mathrm{Col}}\left(N_r\right)$.

Proof. 

By Remark 1 (3), we see that $G=B⋊H$. By Proposition 1, there exists some $h\in G$ such that ${\varphi |}_B$ is a Coleman automorphism of B, where $\varphi =\phi {\sigma }_h$. Let P be a Sylow p-subgroup of G. Thus, we can also assume that ${\varphi |}_{G/B}{=id|}_{G/B}$ and ${\varphi |}_P{=id|}_P$. Let $r\in R_i$ for some i. Since $N_r$ is a direct factor of B, ${\varphi |}_{N_r}$ is a Coleman automorphism of $N_r$.

Then we will show that ${\varphi |}_{N_r}$ is also class-preserving in $N_r$. For every $1\ne f\in N_r$, set $f\left(r\right)=a$ with $a\ne 1$. Keep in mind that $\varphi$ is a class-preserving automorphism of G. Then there exists some $g_0\in G$ such that $f^{\varphi }=f^{g_0}$, where $g_0=b_0{h}_0$, with $b_0\in B,h_0\in H$. Write $b_0={\mathsf{\Pi }}_{t\in R}{b}_t$, with $b_t\in N_t$ as in Remark 1 (4). Therefore,

$$f^{\varphi }=f^{b_0{h}_0}=f^{b_r{h}_0}={\left(f^{b_r}\right)}^{h_0}\in N_r.$$

By Remark 1 (3), it follows that $r{h_0}^{-1}=u^{\prime }r$, with $u\in U_i$. If $b_r\left(r\right)=b\in N$, then $f^{b_r}\left(r\right)=a^b\ne 1$. Keep in mind that ${\tau }_i\left(u^{\prime -1}\right)\in {Aut}_{\mathrm{c}}\left(N\right)$. Therefore, $f^{\varphi }=f^{\prime }\in N_r$, where $f^{\prime }\left(r\right)={\left(a^b\right)}^{{\tau }_i\left(u^{\prime -1}\right)}$, a conjugation of a in N. Write $f^{\prime }\left(r\right)=a^c$ for some $c\in N$ and let $g\in N_r$ satisfy $g\left(r\right)=c$. It follows that $f^{\varphi }=f^{\prime }=f^g$, and the proof is complete. □

Theorem 1.

Let G be a group with a normal subgroup N. Suppose that all central units of $\mathbb{Z}(G/N)$ are trivial and that ${Out}_{\mathrm{Col}}\left(N\right)$ is of odd order. Let P be a Sylow 2-subgroup of G. Then G has the normalizer property if one of the following holds.

(1) 

$Z(P\cap N)\le Z\left(N\right)$ or $Z(P\cap N)\le Z\left(P\right)$;

(2) 

$Z\left(N\right)$ is of odd order and $C_G\left(N\right)\le N$;

(3) 

$Z(P/O_2\left(N\right))\le Z\left(P\right)/O_2\left(N\right)$ and $C_{G/O_2\left(N\right)}(N/O_2\left(N\right))\le N/O_2\left(N\right)$.

Proof. 

Let $\phi \in {Aut}_{\mathbb{Z}}\left(G\right)$ be of 2-power order, and we will show $\phi \in Inn\left(G\right)$. By Proposition 1, we have $\psi =\phi {\sigma }_y\in {Aut}_{\mathrm{Col}}\left(G\right)$ for some $y\in N$ such that

$${\psi |}_{G/N}{=id|}_{G/N},$$

(5)

$${\psi |}_P{=id|}_P.$$

(6)

By Lemma 3 and Remark 2, we assume that $\psi$ is of 2-power order. Hence,

$${\psi |}_N={\sigma }_n{|}_N$$

(7)

with $n\in N$. Since N is normal in G, every Sylow subgroup $N_r$ of N satisfies $N_r^{\psi }\le N$. It follows that $N^{\psi }=N$. By Lemma 8, we may assume that n is a 2-element.

(1)

$Z(P\cap N)\le Z\left(N\right)$ or $Z(P\cap N)\le Z\left(P\right)$.

By (6) and (7), we get ${\sigma }_n{{|}_{P\cap N}=id|}_{P\cap N}$. Notice that $P\cap N$ is a Sylow 2-subgroup of N, and hence, we see that $n\in Z(P\cap N)$. If $Z(P\cap N)\le Z\left(N\right)$, then ${\psi |}_N{=id|}_N$. Lemma 5 implies that $\psi \in Inn\left(G\right)$. If $Z(P\cap N)\le Z\left(P\right)$, then we set $\theta =\psi {\sigma }_{n^{-1}}$. Since $n^{\psi }=n$, $\theta$ is also a 2-element. (5)–(7) yield ${\theta |}_{G/N}{=id|}_{G/N}$, ${\theta |}_P{=id|}_P$, and ${\theta |}_N{=id|}_N$. Again by Lemma 5, we get $\theta \in Inn\left(G\right)$.

(2)

$Z\left(N\right)$ is of odd order and $C_G\left(N\right)\le N$.

By Proposition 1, we have $\varphi =\phi {\sigma }_x$ for some $x\in G$ such that ${\varphi |}_N\in {Aut}_{\mathrm{Col}}\left(N\right)$. We may assume that $\varphi$ is of 2-power order. Since ${Out}_{\mathrm{Col}}\left(N\right)$ is of odd order, we see that ${\varphi |}_N\in Inn\left(N\right)$. Keep in mind that N is normal in G. By Lemma 7 and Remark 2, we assume that ${\varphi |}_N{=id|}_N$ and $\varphi$ is of 2-power order. Since $C_G\left(N\right)\le N$, we obtain ${\varphi |}_{G/N}{=id|}_{G/N}$. Lemma 5 yields ${\varphi |}_{G/O_2(Z\left(N\right))}{=id|}_{G/O_2(Z\left(N\right))}$, and thus $\varphi =id$.

(3)

$Z(P/O_2\left(N\right))\le Z\left(P\right)/O_2\left(N\right)$ and $C_{G/O_2\left(N\right)}(N/O_2\left(N\right))\le N/O_2\left(N\right)$.

If $O_2\left(N\right)=1$, then the conclusion holds by $\left(2\right)$. Therefore, we assume that $O_2\left(N\right)>1$ and consider the factor group $G/O_2\left(N\right)$. Clearly, we see that $O_2(N/O_2\left(N\right))=1$, and it is easy to check that condition $\left(3\right)$ holds for $G/O_2\left(N\right)$. Hence, $G/O_2\left(N\right)$ has the normalizer property. Then there exists some 2-element $g\in G$ such that

$${\psi |}_{G/O_2\left(N\right)}={\sigma }_g{|}_{G/O_2\left(N\right)}.$$

(8)

Again by (6), we obtain $g{O}_2\left(N\right)\in Z(P/O_2\left(N\right))\le Z\left(P\right)/O_2\left(N\right)$. Set $\tau =\psi {\sigma }_{g^{-1}}$. Since $g^{\psi }=g$, we obtain that $\tau$ is of 2-power order. By (8), we see that ${\tau |}_{G/O_2\left(N\right)}{=id|}_{G/O_2\left(N\right)}$. Noticing $g\in Z\left(P\right)$, we see that ${\tau |}_P{=id|}_P$ and ${\tau |}_{O_2\left(N\right)}{=id|}_{O_2\left(N\right)}$. Therefore, Lemma 5 yields the final conclusion. □

When considering generalized twisted wreath products, we can derive a result analogous to Theorem 1 by using Proposition 2.

Theorem 2.

Let $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ be the generalized twisted wreath product of N, with H as in Definition 1. Suppose that all central units of $\mathbb{Z}H$ are trivial, ${\tau }_i\left(U_i\right)\le {Aut}_{\mathrm{c}}\left(N\right)$ for each i, and ${Out}_{\mathrm{c}}\left(N\right)\cap {Out}_{\mathrm{Col}}\left(N\right)$ is of odd order. Suppose that ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$. Then G has the normalizer property if one of the following holds.

(1) 

$Z\left(N\right)$ is of odd order;

(2) 

$Z(P\cap N)\le Z\left(N\right)$, with P a Sylow 2-subgroup of G.

Proof. 

Let $\phi \in {Aut}_{\mathbb{Z}}\left(G\right)$ be of 2-power order, and we will show $\phi \in Inn\left(G\right)$. Write $G=B⋊H$ with B the base group of G.

(1)

$Z\left(N\right)$ is of odd order.

By Proposition 2, there exists some $h\in G$ such that $\left(\phi {\sigma }_h\right){|}_{N_r}$ is a class-preserving Coleman automorphism of $N_r$, where $N_r$ is the direct factor of B as in Remark 1 (4). By Lemma 3 and Remark 2, we may assume that $\varphi =\phi {\sigma }_h$ is of 2-power order, and thus, ${\varphi |}_{N_r}={\sigma }_{x_r}{|}_{N_r}$ for some $x_r\in N_r$. It follows that ${\varphi |}_B={\sigma }_x{|}_B$, with $x={\mathsf{\Pi }}_{r\in R}{x}_r$. Keep in mind that B is normal in G. By Lemma 7, we assume that ${\varphi |}_B{=id|}_B$. Write $G_1=G⋊\langle \varphi \rangle$, and we see that $\left[\right[G,B],\langle \varphi \rangle ]=\left[\right[B,\langle \varphi \rangle ],G]=1$. By the three subgroups’ lemma, it follows that $\left[\right[\langle \varphi \rangle ,G],B]=1$, namely, $g^{-1}{g}^{\varphi }\in C_G\left(B\right)$ for every $g\in G$. Since $C_G\left(B\right)\le B$ by Lemma 1, we obtain ${\varphi |}_{G/B}{=id|}_{G/B}$. It follows from Lemma 5 that ${\varphi |}_{G/O_2(Z\left(B\right))}{=id|}_{G/O_2(Z\left(B\right))}$. Notice that

$$O_2(Z\left(B\right))={\times }_{r\in R}{O}_2(Z\left(N_r\right)).$$

Therefore, $O_2(Z\left(N\right))=1$ yields $\varphi =id\in Inn\left(G\right)$.

(2)

$Z(P\cap N)\le Z\left(N\right)$, with P a Sylow 2-subgroup of G.

Applying Proposition 2, there exists an $h\in G$ such that $\varphi =\phi {\sigma }_h$ satisfies

$${\varphi |}_{G/B}{=id|}_{G/B},$$

(9)

$${\varphi |}_P{=id|}_P$$

(10)

and ${\varphi |}_{N_r}\in {Aut}_{\mathrm{c}}\left(N_r\right)\cap {Aut}_{\mathrm{Col}}\left(N_r\right)$. We assume that $\varphi$ is of 2-power order and ${\varphi |}_B={\sigma }_x{|}_B$ for some $x\in B$. Clearly, $B^{\varphi }=B$. By Lemma 8, we can assume that x is a 2-element. Since $P\cap B$ is a Sylow 2-subgroup of B, ${\varphi |}_{P\cap B}{=id|}_{P\cap B}$ yields

$$x\in Z(P\cap B)={\times }_{r\in R}Z(P\cap N_r)\le {\times }_{r\in R}Z\left(N_r\right)=Z\left(B\right).$$

Thus, we get ${\varphi |}_B{=id|}_B$. Therefore, $\varphi \in Inn\left(G\right)$ by Lemma 5. □

We say that a group G has property $\mathcal{QW}$ if ${Out}_{\mathrm{c}}\left(H\right)\cap {Out}_{\mathrm{Col}}\left(H\right)$ is of odd order for both $H=G$ and $H=G/O_2\left(G\right)$. Then we consider the following theorem.

Theorem 3.

Let $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ be the generalized twisted wreath product of N, with H as in Definition 1. Suppose that N has property $\mathcal{QW}$, $O_2\left(N\right)\le Z\left(N\right)$, and all central units of $\mathbb{Z}H$ are trivial. For each i, suppose that ${\tau }_i\left(u\right)\in {Aut}_{\mathrm{c}}\left(N\right)$ for all $u\in U_i$ and ${\tau }_i\left(u\right)=id$ whenever $u\in U_i$ is a 2-element. If ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$, then G has the normalizer property.

Proof. 

Let $\phi \in {Aut}_{\mathbb{Z}}\left(G\right)$ be of 2-power order, and we have to show $\phi \in Inn\left(G\right)$. Set $G=B⋊H$, with B the base group of G. If $O_2\left(N\right)=1$, then $Z\left(N\right)$ is of odd order. Theorem 2 (1) yields the conclusion. Therefore, assume that $O_2\left(N\right)>1$.

Let P be a Sylow 2-subgroup of G such that $H_2=P\cap H$ is a Sylow 2-subgroup of H. By Lemma 3, Remark 2, and Proposition 1, we may assume that $\phi$ is of 2-power order, satisfying

$${\phi |}_{G/B}=id$$

(11)

and

$${\phi |}_P=id.$$

(12)

If N is a 2-group, then ${\phi |}_B=id$, and therefore, $\phi \in Inn\left(G\right)$ by Lemma 5. Hence, we consider N to not be a 2-group.

Since $O_2\left(B\right)={\times }_{r\in R}{O}_2\left(N_r\right)$, we see that $G/O_2\left(B\right)\cong (N/O_2\left(N\right)){\wr }_{\mathcal{S},\mathcal{T}}H$. Note that $O_2\left(N\right)$ is characteristic in N. Hence, for each $u\in U_i$, the automorphism ${\tau }_i\left(u\right)$ of N naturally induces an automorphism of the quotient group $N/O_2\left(N\right)$. Furthermore, observe that two elements that are conjugate in N remain conjugate under the natural projection onto $N/O_2\left(N\right)$. Thus, ${\tau }_i\left(u\right)$ can be regarded as a class-preserving automorphism of $N/O_2\left(N\right)$. In particular, for any 2-element $u\in U_i$, ${\tau }_i\left(u\right)$ induces the identity automorphism on $N/O_2\left(N\right)$. It follows from $O_2(N/O_2\left(N\right))=1$ that

$${\phi |}_{G/O_2\left(B\right)}={\sigma }_y{|}_{G/O_2\left(B\right)}$$

(13)

for some 2-element $y\in G$. And (12) implies that $y\in P$. By Proposition 2, we say that

$${\phi |}_B={\sigma }_x{|}_B$$

(14)

for some 2-element in B. Set $B_2=B\cap P$. Assume that B is 2-closed, and then ${\phi |}_B=id$. By Lemma 1, (13) yields

$$y{O}_2\left(B\right)\in C_{P/O_2\left(B\right)}(B/O_2\left(B\right))\le B_2/O_2\left(B\right)=1.$$

According to Lemma 5, (12) and (13) imply $\phi \in Inn\left(G\right)$.

Now, consider B to not be 2-closed. It follows that

$${\sigma }_y{|}_{B_2/O_2\left(B\right)}=\phi {{|}_{B_2/O_2\left(B\right)}=id|}_{B_2/O_2\left(B\right)}.$$

(15)

By Lemma 1, we get

$$y{O}_2\left(B\right)\in C_{P/O_2\left(B\right)}(B_2/O_2\left(B\right))\le B_2/O_2\left(B\right),$$

and thus, $y\in B_2$. Then we proceed in four steps.

Step 1.

Let $r\in R_i$ as in Definition 1. If $r{h}_j^{-1}=u_j{r}^{\prime }$ for $h_j\in H_2$ with $j=1,2$, then ${\tau }_i\left(u_1\right)={\tau }_i\left(u_2\right)$.

Clearly, $u_j\in U_i$. Then $u_1^{-1}{u}_2=r^{\prime }\left(h_1{h}_2^{-1}\right)r^{\prime -1}$ is a 2-element in $U_i$. It follows from the assumptions of the theorem that ${\tau }_i\left(u_1^{-1}{u}_2\right)=id$, namely, ${\tau }_i\left(u_1\right)={\tau }_i\left(u_2\right)$.

Step 2.

There is a $z\in B_2$ such that $y{O}_2\left(B\right)=z{O}_2\left(B\right)$ and $[z,H_2]=1$.

Write $y\in B_2$, with $y\left(r\right)=y_r$ for each $r\in R$. Let $h\in H_2$. By (12) and (13), we obtain $y{O}_2\left(B\right)=y{O}_2{\left(B\right)}^{h{O}_2\left(B\right)}$. Therefore, it follows from Remark 1 (3) that

$$y_r{O}_2\left(N\right)=y_{r^{\prime }}^{{\tau }_i\left(u^{\prime -1}\right)}{O}_2\left(N\right)$$

(16)

for all $r\in R_i$ and all i, where $r{h}^{-1}=u^{\prime }{r}^{\prime }$, with $u^{\prime }\in U_i,r^{\prime }\in R_i$.

Notice that $H_2$ acts faithfully on the set $R=R_1\bigsqcup R_2\bigsqcup \cdots \bigsqcup R_k$. Write ${\mathcal{O}}_1,\cdots ,{\mathcal{O}}_l$ to denote all the orbits of the action of $H_2$ on R, and clearly, each ${\mathcal{O}}_s$ is contained in some $R_i$. Now we define an element $z\in B_2$ by using (16). In fact, fix some $r\in {\mathcal{O}}_s$ and set $z\left(r\right)=y_r$. If $r^{\prime }\in {\mathcal{O}}_s$ satisfies $r{h}^{-1}=u^{\prime }{r}^{\prime }$ for some $h\in H_2$ and $u^{\prime }\in U_i$, then by Step 1, we can set $z\left(r^{\prime }\right)=y_r^{{\tau }_i\left(u^{\prime }\right)}$, which does not depend on the choice of h. Then (16) yields $y{O}_2\left(B\right)=z{O}_2\left(B\right)$.

Finally, we show that $[z,H_2]=1$. Let $h^{\prime }\in H_2$ and $r^{\prime }\in {\mathcal{O}}_s\subseteq R_i$. If $r^{\prime }{h}^{\prime -1}=u^{\prime \prime }{r}^{\prime \prime }$ with $u^{\prime \prime }\in U_i$ and $r^{\prime \prime }\in {\mathcal{O}}_s$, then we get $z^h\left(r^{\prime }\right)=z{\left(r^{\prime \prime }\right)}^{{\tau }_i\left(u^{\prime \prime -1}\right)}$. Keep in mind that $r{h}^{-1}=u^{\prime }{r}^{\prime }$, and thus, $r{\left(h^{\prime }h\right)}^{-1}=u^{\prime }{u}^{\prime \prime }{r}^{\prime \prime }$. This implies

$$z^h\left(r^{\prime }\right)=z{\left(r^{\prime \prime }\right)}^{{\tau }_i\left(u^{\prime \prime -1}\right)}=y_r^{{\tau }_i\left(u^{\prime }{u}^{\prime \prime }{u}^{\prime \prime -1}\right)}=y_r^{{\tau }_i\left(u^{\prime }\right)}=z\left(r^{\prime }\right).$$

Step 3.

There is a $u\in O_2\left(B\right)$ such that $x=uz$.

By (12) and (14), we see that $x\in B_2$. According to Step 2, $x{z}^{-1}\in B_2$. By (13) and (14), it follows that

$${\sigma }_x{|}_{B/O_2\left(B\right)}=\phi {{|}_{B/O_2\left(B\right)}={\sigma }_z|}_{B/O_2\left(B\right)},$$

which implies $x{z}^{-1}{O}_2\left(B\right)\in Z(B/O_2\left(B\right))$. Therefore, we obtain

$$x{z}^{-1}{O}_2\left(B\right)\in O_2(B/O_2\left(B\right))=1.$$

This claim is proved.

Step 4.

$\phi \in Inn\left(G\right)$.

It follows from $O_2\left(N\right)\le Z\left(N\right)$ that $O_2\left(B\right)\le Z\left(B\right)$, and thus, $u\in Z\left(B\right)$. By (14) and Step 3, we have

$${\phi |}_B={\sigma }_x{{|}_B={\sigma }_z|}_B.$$

(17)

Let $\varphi =\phi {\sigma }_{z^{-1}}$. Since $z^{\phi }=z$, we see that $\varphi$ is of 2-power order. Thus, (11) and (17) yield ${\varphi |}_{G/B}=id$ and ${\varphi |}_B=id$. Keep in mind that $P=B_2{H}_2$ and $[z,H_2]=1$. And (12) implies that ${\varphi |}_P=id$. Lemma 5 yields $\varphi \in Inn\left(G\right)$. Therefore, $\phi \in Inn\left(G\right)$, and the proof is complete. □

4. The Class-Preserving Coleman Automorphisms of Generalized Twisted Wreath Products

In this section, we study the class-preserving automorphisms and Coleman automorphisms of the generalized twisted wreath products. We first present the following fundamental theorem. This result will enable us to transition from the outer class-preserving automorphism group and outer Coleman automorphism group of a certain factor group to the case of the full group.

Theorem 4.

Let G be a group and $1\ne N⊴G$ be p-closed for some prime p. Let $Q=O_q\left(N\right)$ for some prime $q\in \pi \left(N\right)$. Suppose that $C_G\left(Q\right)\le N$ and that $N/Q{C}_G\left(Q\right)$ is self-centralizing in $G/Q{C}_G\left(Q\right)$. If ${Out}_{\mathrm{Col}}(G/Q)$ or ${Out}_{\mathrm{c}}(G/Q)\cap {Out}_{\mathrm{Col}}(G/Q)$ is a $p^{\prime }$-group, then so is ${Out}_{\mathrm{Col}}\left(G\right)$ or ${Out}_c\left(G\right)\cap {Out}_{\mathrm{Col}}\left(G\right)$, respectively.

Proof. 

As the proofs of the two cases are completely analogous, we restrict our attention to the first case. Assume that ${Out}_{\mathrm{Col}}(G/Q)$ is a $p^{\prime }$-group, and we will prove that ${Out}_{\mathrm{Col}}\left(G\right)$ is a $p^{\prime }$-group as well.

Let $\phi \in {Aut}_{\mathrm{Col}}\left(G\right)$ be of p-power order. It suffices to show that $\phi \in Inn\left(G\right)$. Notice that ${\phi |}_{G/Q}\in {Aut}_{\mathrm{Col}}(G/Q)$ and that ${Out}_{\mathrm{Col}}(G/Q)$ is a $p^{\prime }$-group. It follows that ${\phi |}_{G/Q}\in Inn(G/Q)$. Since Q is normal in G, we see that $Q^{\phi }=Q$. By Lemma 7, we may assume that

$${\phi |}_{G/Q}{=id|}_{G/Q}.$$

(18)

Since $Q\le Q{C}_G\left(Q\right)$, it follows that

$${\phi |}_{G/Q{C}_G\left(Q\right)}{=id|}_{G/Q{C}_G\left(G\right)}.$$

(19)

Then the proof splits into two cases.

Case 1.

$p\ne q$.

Since $\phi$ is a Coleman automorphism, it follows from Lemma 8 that there exists some p-element $x\in G$ such that

$${\phi |}_Q={\sigma }_x{|}_Q.$$

(20)

By Lemma 6, (20) yields

$${\phi |}_{G/Q{C}_G\left(Q\right)}={\sigma }_x{|}_{G/Q{C}_G\left(G\right)}.$$

(21)

Combining (19) and (21), we obtain

$$xQ{C}_G\left(G\right)\in Z(G/Q{C}_G\left(G\right)).$$

(22)

Keep in mind that $N/Q{C}_G\left(Q\right)$ is self-centralizing. It follows that

$$Z(G/Q{C}_G\left(G\right))\le C_{G/Q{C}_G\left(Q\right)}(N/Q{C}_G\left(Q\right))\le N/Q{C}_G\left(Q\right).$$

(23)

Since N is p-closed, we see that the Sylow p-subgroup $O_p\left(N\right)$ of N is contained in $C_G\left(Q\right)$. This implies that $N/Q{C}_G\left(Q\right)$ is a $p^{\prime }$-group. Therefore, (22) and (23) yield $x\in Q{C}_G\left(Q\right)$. Note that $Q{C}_G\left(Q\right)\le N$ is also a p-closed group and that x is of p-power order. It follows that $x\in O_p\left(N\right)$, which implies that $x\in C_G\left(Q\right)$. By (20), we obtain

$${\phi |}_Q{=id|}_Q.$$

(24)

According to Lemma 5, (18), and (24), we get

$${\phi =\phi |}_{G/O_p(Z\left(Q\right))}{=id|}_{G/O_p(Z\left(Q\right))}=id,$$

and the proof is complete in Case 1.

Case 2.

$p=q$.

Since p-group $\langle \phi \rangle$ acts on G, there exists some Sylow p-subgroup P of G such that $P^{\phi }=P$. By Lemma 8, we have

$${\phi |}_P={\sigma }_y{|}_P$$

(25)

for some p-element y. This yields

$${\phi |}_Q={\sigma }_y{|}_Q.$$

(26)

By Lemma 6, we obtain

$${\phi |}_{G/Q{C}_G\left(Q\right)}={\sigma }_y{|}_{G/Q{C}_G\left(G\right)}.$$

(27)

Combining (19) and (27), we see that $yQ{C}_G\left(Q\right)\in N/Q{C}_G\left(Q\right)$. Notice that Q is the normal Sylow p-subgroup of N. Therefore, we get $y\in Q{C}_G\left(Q\right)$, and this yields $y\in Q$.

Set $\psi =\phi {\sigma }_{y^{-1}}$. Then $y^{\phi }=y$ implies that $\psi$ is of p-power order. By (18), (25), and (26), it follows that

$${\psi |}_{G/Q}{=id|}_{G/Q},$$

(28)

$${\psi |}_P{=id|}_P,$$

(29)

and

$${\psi |}_Q={id}_Q.$$

(30)

Therefore, Lemma 5 and (28)–(30) yield $\psi \in Inn\left(G\right)$. Thus, we obtain the final conclusion, namely, $\phi \in Inn\left(G\right)$. □

By using the method established by Petit Lobão and Sehgal in [7], we obtain the following Lemma on generalized twisted wreath products.

Lemma 9.

Let N be a nontrivial nilpotent group and H a group. Denote by $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ the generalized twisted wreath product of N, with H as in Definition 1, and B the base group. Suppose that ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$. If $\phi \in {Aut}_{\mathrm{c}}\left(G\right)\cap {Aut}_{\mathrm{Col}}\left(G\right)$, then there exists some $x\in G$ such that ${\phi |}_B={\sigma }_x{|}_B$.

Proof. 

Since N is nilpotent, so is B, and hence, we can write $B=P_1\times \cdots \times P_s$, where the $P_i$ are Sylow subgroups of B. Keep in mind that $\phi \in {Aut}_{\mathrm{Col}}\left(G\right)$. For each $P_i$, there exist $b_i\in P_i$ and $h_i\in H$ such that ${\phi |}_{P_i}={\sigma }_{b_i{h}_i}{|}_P$. Then we will show that all $h_i$ are equal.

Let $x\in Z\left(P_i\right)$ and $y\in Z\left(P_j\right)$. There exists some $h_{x,y}\in H$ such that ${\left(xy\right)}^{\phi }={\left(xy\right)}^{h_{x,y}}=x^{h_{x,y}}{y}^{h_{x,y}}$. On the other hand, $x^{\phi }=x^{h_i}$ and $y^{\phi }=y^{h_j}$. Therefore,

$$x^{h_{x,y}}=x^{\phi }=x^{h_i},$$

(31)

$$y^{h_{x,y}}=y^{\phi }=y^{h_j}.$$

(32)

According to Remark 1 (4), $B={\times }_{r\in R}{B}_r$. Fix an $r\in R_s$. Now choose $f,g\in B_r$ such that $f\in Z\left(P_i\right)$ with $f\left(r\right)\ne 1$ and $g\in Z\left(P_j\right)$ with $g\left(r\right)\ne 1$. According to (31) and (32), assume that $r^{\prime }{h}_{f,g}^{-1}=u^{\prime }r$, $r^{\prime \prime }{h}_i^{-1}=u^{\prime \prime }r$ and $r^{\prime \prime \prime }{h}_j^{-1}=u^{\prime \prime \prime }r$, where $u^{\prime },u^{\prime \prime },u^{\prime \prime \prime }\in U_s$ and $r^{\prime },r^{\prime \prime },r^{\prime \prime \prime }\in R_s$. By Remark 1 (3),

$$f^{h_{f,g}}\left(r^{\prime }\right)=f^{h_i}\left(r^{\prime \prime }\right)\ne 1,$$

(33)

$$g^{h_{f,g}}\left(r^{\prime }\right)=g^{h_j}\left(r^{\prime \prime \prime }\right)\ne 1.$$

(34)

Since $f,g\in B_r$, they are nonzero only at the position r. Consequently, each of the functions $f^{h_{f,g}}$, $g^{h_{f,g}}$, $f^{h_i}$, and $g^{h_j}$ is also nonzero at exactly one position. Notice that $f^{h_{f,g}}=f^{h_i}$ and $g^{h_{f,g}}=g^{h_j}$. We deduce that $r^{\prime }=r^{\prime \prime }=r^{\prime \prime \prime }$ by (33) and (34). Therefore, $h_i{h}_j^{-1}\in {\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$, namely, $h_i=h_j$. Denote by $h=h_1=\cdots =h_s$ and write $x=a_1{a}_2\cdots a_{s}h$. It follows that ${\phi |}_B={\sigma }_x{|}_B$. □

In the case where N is a nilpotent group, we obtain the following theorem.

Theorem 5.

Let N be a nontrivial nilpotent group and H a group. Denote by $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ the generalized twisted wreath product of N, with H as in Definition 1. If ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$, then ${Out}_{\mathrm{c}}\left(G\right)\cap {Out}_{\mathrm{Col}}\left(G\right)=1$.

Proof. 

Let $\phi \in {Aut}_{\mathrm{c}}\left(G\right)\cap {Aut}_{\mathrm{Col}}\left(G\right)$ be of p-power order for some prime $p\in \pi \left(G\right)$. Denote by B the base group of G. By Lemmas 7 and 9, we may assume

$${\phi |}_B{=id|}_B.$$

(35)

Lemma 1 yields

$${\phi |}_{G/B}{=id|}_{G/B}.$$

(36)

If $p\notin \pi \left(B\right)$, then it follows from (35), (36), and Lemma 5 that $\phi =id$, and the proof is complete. Hence, we may assume $p\in \pi \left(B\right)$. Let P be a Sylow p-subgroup of G such that $P^{\phi }=P$. By Lemma 8, there exists some p-element $g\in G$ such that

$${\phi |}_P={\sigma }_g{|}_P.$$

(37)

Clearly, $g\in P$. By (35), (37), and Lemma 1, we obtain $g\in C_G(P\cap B)\le P\cap B$, namely, $g\in Z\left(B\right)$. Write $\psi =\phi {\sigma }_{g^{-1}}$, and we see that $\psi$ is of p-power order. It follows from Lemma 5 and (35)–(37) that $\tau \in Inn\left(G\right)$. Hence, the proof is completed. □

We now investigate Coleman automorphisms of generalized twisted wreath products constructed from a nilpotent group N. Firstly, we recall the following lemma.

Lemma 10

([13] Theorem 1.1). Let N be a nontrivial nilpotent normal subgroup of a group G. Assume that $Z(G/N)=1$ and that the centralizer of every Sylow subgroup of N in G is contained in N. Then ${Out}_{\mathrm{Col}}\left(G\right)=1$.

Theorem 6.

Let N be a nontrivial nilpotent group and H a group with $Z\left(H\right)=1$. Denote by $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ the generalized twisted wreath product of N, with H as in Definition 1. If ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$, then ${Out}_{\mathrm{Col}}\left(G\right)=1$.

Proof. 

Denote by B the base group of G. It follows that $Z(G/B)\cong Z\left(H\right)=1$. By Lemma 1, the centralizer of every Sylow subgroup of B in G is contained in B. Lemma 10 implies that ${Out}_{\mathrm{Col}}\left(G\right)=1$. □

Before we consider the p-closed groups N, we need the following Lemma by Isaacs and Li in [6].

Lemma 11

([6] Lemma 2.4). Let G be a solvable group. Then G is nilpotent if and only if $G=P{C}_G\left(P\right)$ for every $P\in \mathrm{Syl}(F(G\left)\right)$.

Now we are ready to prove the following theorem.

Theorem 7.

Let N be a nontrivial p-closed solvable group, with p a prime, and let H be a group. Denote by $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ the generalized twisted wreath product of N, with H as in Definition 1. Suppose that ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$. Then ${Out}_{\mathrm{c}}\left(G\right)\cap {Out}_{\mathrm{Col}}\left(G\right)$ is a $p^{\prime }$-group. Furthermore, if $Z\left(H\right)=1$, then ${Out}_{\mathrm{Col}}\left(G\right)$ is a $p^{\prime }$-group.

Proof. 

By Theorems 5 and 6, we may assume that N is not nilpotent. It follows from Lemma 11 that there exists a Sylow q-subgroup S of $F\left(N\right)$ such that $S{C}_N\left(S\right)<N$. Clearly, $S{C}_N\left(S\right)$ is characteristic in N. Adopting the symbols in Definition 1 and Remark 1, set $S_r=\{f\mid f\left(r\right)\in S\mathrm{and}f\left(s\right)=1ifs\ne r\}$ and $Q={\times }_{r\in R}{S}_r$. Clearly, we see that $Q=O_q\left(B\right)$, where B is the base group. Lemma 1 yields $C_G\left(Q\right)\le B$. It is easy to check that

$$G/Q{C}_G\left(Q\right)=G/Q{C}_B\left(Q\right)\cong (N/S{C}_N\left(S\right)){\wr }_{\mathcal{S},\mathcal{T}}H.$$

(38)

Note that both the properties of being p-closed and solvable are inherited by quotient groups, and hence, $N/S{C}_N\left(S\right)$ is a p-closed solvable group. Since $1<|N/S{C}_N\left(S\right)|<|N|$, we see that ${Out}_{\mathrm{c}}(G/Q{C}_G\left(Q\right))\cap {Out}_{\mathrm{Col}}(G/Q{C}_G\left(Q\right))$ is a $p^{\prime }$-group by induction. The induction starts from the base case, where the base group is nilpotent, which is already established by Theorem 5. It follows from Theorem 4 that ${Out}_{\mathrm{c}}\left(G\right)\cap {Out}_{\mathrm{Col}}\left(G\right)$ is also a $p^{\prime }$-group, and the proof is complete in this case.

Now suppose that $Z\left(H\right)=1$. Similarly, by induction and Theorem 6, we obtain that ${Out}_{\mathrm{Col}}(G/Q{C}_G\left(Q\right))$ is a $p^{\prime }$-group. Theorem 4 implies that ${Out}_{\mathrm{Col}}\left(G\right)$ is a $p^{\prime }$-group. □

Notice that 2-closed groups are all solvable. Therefore, Theorem 7 yields the following corollary.

Corollary 1.

Let N be a nontrivial 2-closed group and H be a group. Denote by $G=N{\wr }_{\mathcal{S},\mathcal{T}}H$ the generalized twisted wreath product of N, with H as in Definition 1. Suppose that ${\bigcap }_{i=1}^k{\mathrm{Core}}_H\left(U_i\right)=1$. Then ${Out}_{\mathrm{c}}\left(G\right)\cap {Out}_{\mathrm{Col}}\left(G\right)$ is a $2^{\prime }$-group. Furthermore, if $Z\left(H\right)=1$, then ${Out}_{\mathrm{Col}}\left(G\right)$ is a $2^{\prime }$-group. In particular, G has the normalizer property.

5. Discussion

We have introduced the generalized twisted wreath products that encompass the standard wreath product, the regular wreath product, and the twisted wreath product as special cases. Our results are thus widely applicable and naturally generalize several previous works, such as those in [5,6,8,14,15].

In conclusion, the concept of symmetry manifests in multiple facets of our work. Firstly, the generalized twisted wreath product itself is a symmetrical object, built from a group N and the symmetrical action of a group H on a set. Secondly, the automorphism groups we study—specifically the class-preserving and Coleman automorphisms—are fundamental measures of a group’s internal symmetry. Furthermore, the normalizer property $N_{U\left(\mathbb{Z}G\right)}\left(G\right)=G·C_{U\left(\mathbb{Z}G\right)}\left(G\right)$ reveals a profound symmetry property: it ensures that all symmetries of G within the unit group of its integral group ring are intrinsic, arising only from G itself and its centralizer. By providing a unified analysis of these automorphisms and establishing the normalizer property for a broad class of groups, our results contribute to a deeper understanding of symmetry preservation in algebraic structures.