# Coleman Automorphisms of Some Metabelian Groups

Authors: Jingjing Hai; Yanxia Li; Xian Ling
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, the approach is through the projective limits, and we obtain some characterizations of the Coleman automorphism groups of a split metabelian group and a generalized dicyclic group. An application of our results is a generalization of the result of Z. Li and Y. Li.


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## Body

## 1. Introduction

Throughout this study, all groups considered are finite andAutF denotes the automorphism group of a finite group F, and we use conjx to denote the inner automorphism g↦gx on F. Then, we have the following two subgroups of AutF.AutColF consists of those ρ∈AutF, such that for any prime q∈πF and any Sylow q-subgroup Q∈SylqF, there exists a u∈F, such that ρ|Q=conju|Q.AutℤF=σv∈AutF|xσv=v−1xv,v∈NUℤFF,x∈F, where ℤF is the integral group ring, and NUℤFF is the normalizer of F in the unit group UℤF of ℤF. We set OutColF=AutColF/InnF, OutℤF=AutℤF/InnF.Recall that Coleman automorphisms come up in the study of the normalizer problem for integral group rings of finite groups, i.e., doesNUℤFF=F⋅ZUℤF hold, where ZUℤF is the center of UℤF. It is easy to see that NUℤFF=F⋅ZUℤF is equivalent to AutℤF=InnF. Coleman’s lemma [1] implies that OutℤF≤OutColF. In [2], Krempa proved that OutℤF is a 2-group. Thus, if we show that OutColF is a group of odd order or AutColF=InnF under some conditions, then AutℤF=InnF, that is, the normalizer problem holds for F. Related results on this subject can be found in [3–9].The purpose of this study is to determine the structure of the Coleman automorphism groups of some metabelian groups. Li and Li [10] obtained a characterization of AutColD of a generalized dihedral group D. In this study, the Coleman automorphism group of a split metabelian group is characterized by using the projection limit property; this result is an extension and alternative proof of Theorem 1.2 in [10]. We also give the structure of AutColG of a generalized dicyclic group G. Our notation is mostly standard, refer to [2, 11, 12].


## 2. Preliminaries

Definition 1.
LetF be a finite group, and let Mi⊲¯F. Put Fi=F/Mi,Fij=F/MiMj, and assume that πij:Fi⟶Fij is the natural homomorphisms, where 1≤i,j≤s,s is a natural number. Then, the projective limit of the quotients Fi with respect to the natural maps πij is the subgroup(1)F^=Projlim1≤i,j≤sFi,πij=h1,h2,…,hs∈∏i=1sFi|hiπij=hjπji,1≤i,j≤s,of the direct product of the quotients Fi ([13], Page 115, Definitions).Lemma 1 (See [14]).
LetF be a finite group, and let Mi⊲¯Fi=1,2,…,s. Assume that(1)
∩i=1sMi=1(2)
For any primeq∈πF, there exists a i, such that q,Mi=1. Then, F≅F^=Projlim1≤i,j≤sFi,πij.Lemma 2 (See [13]).
LetF be a finite solvable group. Then,(2)AutColF=ρ∈AutFρF/Op′F∈InnF/Op′F,p∈πF,where Op′F denotes the largest normal p′-subgroup of F.Lemma 3.
LetG be a finite solvable group. Put Gi=G/Opi′G,Gij=G/Opi′GOpi′G, where πG=p1,…,pn. Then,(1)
The natural mapsπij:Gi⟶Gij induce maps πij∗:Gi/ζGi⟶Gij/ζGij, where ζGi is the center of Gi and ζGij is the center of Gij, respectively.(2)
AutColG≅Projlim1≤i,j≤nGi/ζGi,πij∗≅Projlim1≤i≤nInnGiProof.
(1)
Set(3)πij∗:GiζGi⟶GijζGij,giζGi↦giπijζGij.It easily is verified thatπij∗ is a surjective homomorphism(2)
By Lemma1, we obtain G≅Projlim1≤i,j≤nGi,πij. For any gi∈Gi,gj∈Gj, then Definition 1 implies that giπij=gjπji. Set(4)βi:GiζGi⟶InnGi:Gi⟶Gi,giζGi↦σgi:xi↦xigi,βij:GijζGij⟶InnGij:Gij⟶Gij,gijζGij↦σgij:xij↦xijgij.
Then,βi,βij are the isomorphisms. Assume that(5)θij:InnGi⟶InnGij,σgi↦σgiπij.
It is clear thatθij=βi−1πij∗βij are the surjective homomorphisms. First, we show that(6)Projlim1≤i,j≤nGiζGi,πij∗≅Projlim1≤i≤nInnGi.
In fact,(7)Projlim1≤i,j≤nGiζGi,πij∗=g1ζG1,…,gnζGn∈∏i=1nGiζGi|giζGiπij∗=gjζGjπji∗,1≤i,j≤n.
On the other hand,(8)Projlim1≤i,j≤nInnGi,θij=σg1,…,σgn∈∏i=1nInnGi|σgiθij=σgjθji,1≤i,j≤n.
Let(9)φ:Projlim1≤i,j≤nGiζGi,πij∗⟶Projlim1≤i,j≤nInnGi,θij,g1ζG1,…,gnζGn↦σg1,…,σgn.
It easily is verified thatφ is an isomorphism.
Next, we shall show that(10)AutColG≅Projlim1≤i,j≤nInnGi,θij.
SinceG≅G^=Projlim1≤i,j≤nGi,πij, so it remains to show that(11)AutColG^=Projlim1≤i,j≤nInnGi,θij.
By Definition1 and Lemma 1, G^=G1×⋯×Gn. Thus, for any x=x1,…,xn∈G^ with xi∈Gi,1≤i≤n, σg1,…,σgn∈Projlim1≤i,j≤nInnGi,θij, write(12)x1,…,xnσg1,…,σgn=x1σg1,…,xnσgn=x1g1,…,xngn.
It follows that(13)gi−1xigiπij=gi−1πijxiπijgiπij=gj−1πjixjπjigjπji=gj−1xjgjπji,that is, x1g1,…,xngn∈G^. Moreover, it easily is verified that σg1,…,σgn∈AutG^. By Lemma 2, we obtain σg1,…,σgn∈AutColG^, i.e., Projlim1≤i,j≤nInnGi,θij≤AutColG^.
Conversely, suppose thatσ∈AutColG^, then by Lemma 2, σ|Gi∈InnGi. Write σgi=σ|Gi. Consequently,(14)σ=σg1,…,σgn∈∏i=1nInnGi.
For anyxij∈Gij with 1≤i,j≤n, we have xijσgiθij=xijσgiπij=xijσgjπji=xijσgjθji; it follows that σgiθij=σgjθji, that is, σg1,…,σgn∈Projlim1≤i,j≤nInnGi,θij, which implies that AutColG^≤Projlim1≤i,j≤nInnGi,θij. Hence,(15)AutColG≅Projlim1≤i,j≤nGiζGi,πij∗≅Projlim1≤i≤nInnGi.


## 3. Proof of the Theorems

Theorem 1.
LetG=A⋊B be a split metabelian group, where ab=a−1 for every a∈A,b∈B, A2∈Syl2A, and expA2 denotes the exponent of A2. Then, if πG=1, then AutColG=InnG, and if πG≥2, then AutColG≅InnG⋊BπG−2,expA2≤2AutColG≅InnG⋊BπG−1,expA2>2.Proof.
(1)
IfπG=1, then G is a p-group and AutColG=InnG(2)
IfπG≥2, let A=A2×Ap1×⋯×Aps, where p1,p2,…,ps are distinct odd primes and Api∈SylpiA,i=1,2,…,sCase 1:expA2=1SinceG=Ap1×⋯×Aps⋊B, it follows that ζG=1,(16)O2′G=Ap1×⋯×Aps,Opi′G=Ap1×⋯×Api−1×Api+1⋯×Apsi=1,2,…,s.WriteG1=G/O2′G,G2=G/Op1′G,…,Gs+1=G/Ops′G, then G1≅B,G2≅Ap1⋊B,…,Gs+1≅Aps⋊B. Thus, ζG1≅B,ζG2=1,…,ζGs+1=1.By Lemma3,(17)AutColG≅Projlim1≤i≤s+1InnGi=InnG1×InnG2×⋯×InnGs+1≅Ap1⋊B×⋯×Aps⋊B≅InnG⋊BπG−2.Case 2:expA2=2SinceG=A2×Ap1×⋯×Aps⋊B, expA2=2. It follows that ζG=A2,(18)O2′G=Ap1×⋯×Aps,Opi′G=A2×Ap1×⋯×Api−1×Api+1⋯×Apsi=1,…,s.WriteG1=G/O2′G,G2=G/Op1′G,…,Gs+1=G/Ops′G. Thus, G1≅A2×B,G2≅Ap1⋊B,…,Gs+1≅Aps⋊B, which implies that ζG1≅A2×B,ζG2=1,…,ζGs+1=1. By Lemma 3,(19)AutColG≅Projlim1≤i≤s+1InnGi=InnG1×InnG2×⋯×InnGs+1≅Ap1⋊B×⋯×Aps⋊B≅InnG⋊BπG−2.Case 3:expA2>2
LetA2=C2m1×C2m2×⋯×C2mt, where m1≥m2≥⋯≥mt,t≥1. Since G=A2×Ap1×⋯×Aps⋊B, then ζG=C2×⋯×C2 is of order 2t. Thus,(20)O2′G=Ap1×⋯×Aps,Opi′G=A2×Ap1×⋯×Api−1×Api+1⋯×Apsi=1,…,s.
WriteG1=G/O2′G,G2=G/Op1′G,…,Gs+1=G/Ops′G, we have G1≅A2⋊B,G2≅Ap1⋊B,…,Gs+1≅Aps⋊B,which implies that ζG1≅C2×⋯×C2 is of order 2t, ζG2=1,…,ζGs+1=1. By Lemma 3,(21)AutColG≅Projlim1≤i≤s+1InnGi=InnG1×InnG2×⋯×InnGs+1≅C2m1−1×C2m2−1×⋯×C2mt−1⋊B×Ap1⋊B×⋯×Aps⋊B≅InnG⋊BπG−1.
LetA be an abelian group and γ be an involution. Write DihA,γ=A,γ|γ2=1,aγ=a−1,a∈A. DihA,γ is called a generalized dihedral group. In particular, if A is cyclic, then DihA,γ is a dihedral group.Corollary 1.
LetG=DihA,γ=A⋊γ, A2∈Syl2A. Then, if πG=1, then AutColG=InnG; and if πG≥2, then AutColG≅InnG⋊C2πG−2,expA2≤2AutColG≅InnG⋊C2πG−1,expA2>2.Corollary 2.
LetG=DihA,γ=A⋊γ, A2∈Syl2A. Then, OutColG=1 if and only if either πG=1 or πG=2 and expA2=2.LetE be an abelian group of even order and expE>2, and let v∈E,ov=2. Write DicE,v,u=E,u|u2=v,xu=x−1,x∈E. DicE,v,u is called a generalized dicyclic group. In particular, if E is cyclic, then DicE,v,u is a generalized quaternion group.Theorem 2.
LetG=DicE,v,u, E2∈Syl2E. Then, if πG=1, then AutColG=InnG; and if πG≥2, then AutColG≅InnG⋊C2πG−2,expE2=2AutColG≅InnG⋊C2πG−1,expE2>2.Proof.
(1)
IfπG=1, then G is a 2-group and AutColG=InnG.(2)
IfπG≥2, let E=E2×Ep1×⋯×Eps, where p1,p2,…,ps are the distinct odd primes, Epi∈SylpiE,i=1,2,…,s. Then, πG=2,p1,…,ps.Case 1:expE2=2SinceE2 is an elementary abelian 2-group. It follows that ζG=E2,(22)O2′G=Ep1×⋯×Eps,Opi′G=E2×Ep1×⋯×Epi−1×Epi+1⋯×Epsi=1,…,s.WriteG1=G/O2′G,G2=G/Op1′G,…,Gs+1=G/Ops′G. Thus, G1≅E2×C2,G2≅Ep1⋊C2,…,Gs+1≅Eps⋊C2, which implies that ζG1≅E2×C2,ζG2=1,…,ζGs+1=1. By Lemma 3,(23)AutColG≅Projlim1≤i≤s+1InnGi=InnG1×InnG2×⋯×InnGs+1≅Ep1⋊C2×⋯×Eps⋊C2≅InnG⋊C2πG−2.Case 2:expE2>2
LetE2=C2m1×C2m2×⋯×C2mt, where m1≥m2≥⋯≥mt,t≥1. Thus, ζG=C2×⋯×C2 is of order 2t, and(24)O2′G=Ep1×⋯×Eps,Opi′G=E2×Ep1×⋯×Epi−1×Epi+1⋯×Epsi=1,…,s.
WriteG1=G/O2′G,G2=G/Op1′G,…,Gs+1=G/Ops′G, and we have that(25)G1≅E2u,G2≅Ep1⋊C2,…,Gs+1≅Eps⋊C2.
It follows thatζG1≅C2×⋯×C2 is of order 2t, ζG2=1,…,ζGs+1=1. By Lemma 3,(26)AutColG≅Projlim1≤i≤s+1InnGi=InnG1×InnG2×⋯×InnGs+1≅C2m1−1×C2m2−1×⋯×C2mt−1⋊C2×Ep1⋊C2×⋯×Eps⋊C2≅InnG⋊C2πG−1.Corollary 3.
LetG=DicE,v,u, E2∈Syl2E. Then, if πG=1, then OutColG=1; if πG≥2, then OutColG≅C2πG−2,expE2=2OutColG≅C2πG−1,expE2>2.Corollary 4.
LetG=DicE,v,u, E2∈Syl2E. Then, OutColG=1 if and only if either πG=1 or πG=2 and expE2=2.



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