Normalizer property of finite groups with almost simple subgroups

1.   Introduction

Let $F$ be a finite group and ${\rm Aut}(F)$ be its automorphism group. We use $\mathbb{Z}F$ to denote the integral group ring of $F$. The normalizer problem (see ^[1], problem 43) asks whether ${\rm N}_{{\rm U}({\mathbb{Z}}F)}(F) = F{\rm \zeta}({\rm U}({\mathbb{Z}}F))$, where ${\rm N}_{{\rm U}({\mathbb{Z}}F)}(F)$ is normalizer of $F$ in the unit group ${\rm U}({\mathbb{Z}}F)$, ${\rm \zeta}({\rm U}({\mathbb{Z}}F))$ is the center of ${\rm U}({\mathbb{Z}}F)$. Write ${\rm Aut_{\mathbb{Z}}}(F) = \{\sigma_{v}\in {\rm Aut}(F)\mid x^{\sigma_{v}} = v^{-1}xv, v\in {\rm N}_{{\rm U}({\mathbb{Z}}F)}(F), x\in F\}$, then ${\rm Aut_{\mathbb{Z}}}(F)$ is a subgroup of ${\rm Aut}(F)$. Set ${\rm Out_{\mathbb{Z}}}(F) = {\rm Aut_{\mathbb{Z}}}(F)/{\rm Inn}(F)$. Jackowski and Marciniak ^[2] proved that ${\rm N}_{{\rm U}({\mathbb{Z}}F)}(F) = F{\rm \zeta}({\rm U}({\mathbb{Z}}F))$ is equivalent to ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$. Thus, we call that $F$ has the normalizer property provided that ${\rm Out_{\mathbb{Z}}}(F) = 1$.

Hertweck and Kimmerle ^[3] introduced the Coleman automorphism, i.e., a $\varphi\in {\rm Aut}(F)$ is said to be Coleman automorphism if for any $q\in\pi(F)$ and any $Q\in {\rm Syl}_{q}(F)$, there exists a $h\in F$ with $\varphi|_{Q} = {\rm conj}(h)|_{Q}$. Denoted by ${\rm Aut_{Col}}(F)$ the Coleman automorphism group of $F$. Write ${\rm Out_{Col}}(F) = {\rm Aut_{Col}}(F)/{\rm Inn}(F)$. In ^[4], Gross introduced the $q$-central automorphism, i.e., a $\theta\in {\rm Aut}(F)$ is called a $q$-central if there exists a $q\in\pi(F)$ and some $Q\in {\rm Syl}_{q}(F)$ such that $\theta|_{Q} = id|_{Q}$. Obviously, modifying the Coleman automorphism with an inner automorphism, then the Coleman automorphism of $F$ is $q$-central for any $q\in \pi(F)$.

Coleman automorphisms come up in the study of the normalizer problem. By Coleman's lemma ^[5] and Krempa's result ^[1], we only show that ${\rm Out_{Col}}(F) = 1$ or ${\rm Out_{Col}}(F)$ is a $2'$-group, then ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$. For example, let $F$ be a simple group or a nilpotent group. Then ${\rm Out_{Col}}(F) = 1$(see ^[3]). Related results on this subject can be found in ^[6,7,8,9,10].

The purpose of this paper is to investigate normalizer property of finite groups with almost simple subgroups. Recall that a finite group $A$ is called to be an almost simple group provided that there exists some non-abelian simple group $S$ with $S\leq A \leq {\rm Aut}(S)$. Note that Van Antwerpen ^[10] gave a group $C_{15}\rtimes C_{2}$ for which ${\rm Out_{Col}}(C_{15}\rtimes C_{2})\cong C_{2}$. In this paper, we consider that $F$ is an almost simple group by a simple group or an odd order group by an almost simple group. We shall show that ${\rm Out_{Col}}(F) = 1$ or ${\rm Out_{Col}}(F)$ is of odd order. In particular, these groups have the normalizer property. Our notation is standard, refer to ^[1,3,11].

2.   Preliminaries

Lemma 2.1. ^[3] Assume that $S$ is a simple group. Then there is $q\in\pi(S)$ such that every $q$-central automorphisms of $S$ is inner.

Lemma 2.2. Let $S$ be a non-abelian simple group. Then $C_{{\rm Aut}(S)}({\rm Inn}(S)) = 1$.

Proof. By hypothesis, then ${\rm \zeta}(S) = 1$ and $S\simeq {\rm Inn}(S)$. Set $\sigma:S\rightarrow {\rm Inn}(S)$ is an isomorphism. Thus, for any $\theta \in C_{{\rm Aut}(S)}({\rm Inn}(S)), g\in G$, we obtain $\theta^{-1}\sigma (g)\theta = \sigma (g)$, that is, $\sigma (g^{\theta}) = \sigma (g)$. It follows that $g^{\theta} = g$, which implies that $\theta = 1$. Hence $C_{{\rm Aut}(S)}({\rm Inn}(S)) = 1$.

Lemma 2.3. Let $J\leq F$. Then $C_{F}(J) = 1$ if and only if ${\rm \zeta}(H) = 1$ for every $H$ such that $J\leq H\leq F$.

Proof. The assertion is obvious.

Lemma 2.4. Assume that $A$ is an almost simple group. Then ${\rm \zeta}(A) = 1$.

Proof. By Lemma 2.2 and Lemma 2.3, the conclusion holds.

Lemma 2.5. ^[6] Let $\rho\in {\rm Aut}(F)$ be of $p$-power order and $E\leq F$, where $p\in \pi(F)$. If $\rho|_{E} = {\rm conj}(h)|_{E}$ for some $h\in F$, then there exists a $\delta\in {\rm Inn}(F)$ such that $\rho\delta|_{E} = id|_{E}$ and $o(\rho\delta) = p^{i}$, where $i$ is positive integer.

Lemma 2.6. ^[3] Let $\rho\in {\rm Aut_{Col}}(F)$ and $M\unlhd F$. Then

(1) $\rho|_{M}\in {\rm Aut}(M)$,

(2) $\rho|_{F/M}\in {\rm Aut_{Col}}(F/M)$.

Lemma 2.7. Assume that $A$ is an almost simple group. Then ${\rm Aut_{Col}}(A) = {\rm Inn}(A)$.

Proof. Let $\rho \in {\rm Aut_{Col}}(A)$. We shall prove that $\rho\in {\rm Inn}(A)$. By hypothesis $S\leq A \leq {\rm Aut}(S)$. It follows from Lemma 2.1 that there exists some $q \in \pi(S)$ such that every $q$-central automorphism of $S$ is inner. Let $Q\in {\rm Syl_{q}}(A)$. Since $\rho\in {\rm Aut_{Col}}(A)$, then $\rho|_{Q} = {\rm conj}(a)|_{Q}$ for some $a\in A$. In general, we may suppose that $\rho|_{Q} = id|_{Q}$ by Lemma 2.5. Write $R = Q\cap S$, hence $R\in {\rm Syl_{q}}(S)$ and $\rho|_{R} = id|_{R}$. Note that $S\unlhd A$, by Lemma 2.6(1), we obtain that $\rho|_{S}$ is $q$-central. Hence, $\rho|_{S}\in {\rm Inn}(S)$, i.e., there exists a $g\in S$ with $\rho|_{S} = {\rm conj}(g)|_{S}$. Write $\eta = \rho {\rm conj}(g^{-1})$, then $\eta|_{S} = id|_{S}$. By Lemma 2.2 and $S$ identifies with ${\rm Inn}(S)$, we obtain that $C_{A}(S) = C_{{\rm Aut}(S)}(S)\cap A = 1$. Thus, for any $y\in A$ and $x\in S$, we have $(y^{-1}xy)^{\eta} = (y^{-1})^{\eta}xy^{\eta} = y^{-1}xy$, which implies that $y^{\eta}y^{-1}\in C_{A}(S) = 1$. Hence, $\eta = id$, i.e., $\rho\in {\rm Inn}(A)$.

Lemma 2.8. ^[12] Let $\rho\in {\rm Aut}(F)$ be of $p$-power order. If there exists some $H\lhd F$ such that $\rho|_{H} = id|_{H}$ and $\rho|_{F/H} = id|_{F/H}$, then $\rho|_{F/O_{p}({\rm\zeta}(H))} = id|_{F/O_{p}({\rm\zeta}(H))}$. Assume further that there exists a $T\in {\rm Syl_{p}}(F)$ such that $\rho|_{T} = id|_{T}$. Then $\rho\in {\rm Inn}(F)$.

Lemma 2.9. ^[6] Let $\rho\in {\rm Aut}(F)$ be of $p$-power order, and let $H\lhd F$ with $H^{\rho} = H$. Assume further that $\rho|_{F/H}\in {\rm Inn}(F/H)$. Then there is $\tau\in {\rm Inn}(F)$ such that $\rho\tau|_{F/H} = id|_{F/H}$ and $o(\rho\tau) = p^{j}$, where $j$ is positive integer.

Lemma 2.10. ^[3] Let $\rho\in {\rm Aut}(F)$ and $H\lhd F$ with $H^{\rho} = H$, and assume that $Q\in {\rm Syl}(H)$. If $\rho|_{Q} = {\rm conj}(g)|_{Q}$ for some $g\in F$, then $K = HC_{F}(Q)\unlhd F$ and $K^{\rho} = K$. Moreover, $\rho|_{F/K} = {\rm conj}(g)|_{F/K}$.

Lemma 2.11. ^[3] Let $M$ be a $2'$-group. Then ${\rm Out_{Col}}(M)$ is also a $2'$-group.

3.   Proof of The Theorems

Theorem 3.1. Let $A$ be an almost simple normal subgroup of $F$. If $F/A$ is an abelian group, then ${\rm Out_{Col}}(F) = 1$. In particular, ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$.

Proof. Let $\varphi\in {\rm Aut_{Col}}(F)$ and let $\varphi$ be of $p$-power order, where $p\in \pi(F)$. We shall prove that $\varphi\in {\rm Inn}(F)$. By hypothesis $A$ is almost simple, then $S\leq A \leq {\rm Aut}(S)$. Now, we show that $\varphi|_{S}$ is a $q$-central, where $q \in \pi(S)$ and $S$ is non-abelian simple. Let $Q\in {\rm Syl_{q}}(A)$, then there exists some $T\in {\rm Syl_{q}}(F)$ such that $Q\leq T$. Note that $\varphi \in {\rm Aut_{Col}}(F)$, thus $\varphi|_{T} = {\rm conj}(g)|_{T}$ for some $g\in F$. In general, we suppose that $\varphi|_{T} = id|_{T}$ by Lemma 2.5. Write $R = Q\cap S$, hence $R\in {\rm Syl_{q}}(S)$ and $\varphi|_{R} = id|_{R}$. By Lemma 2.6(1), we obtain that $\varphi|_{A}$ is an automorphism of $A$. Denote by $R^{S}$ the normal closure of $R$ in $S$. Since $S$ is non-abelian simple, then $R^{S} = S$. Note that $R^{S} = <s^{-1}rs: s\in S, r\in R>$ and $S\unlhd A$. Hence, for any $s\in S, r\in R$, $(s^{-1}rs)^{\varphi} = (s^{\varphi})^{-1}rs^{\varphi}\in S$, which implies that $\varphi|_{S}\in {\rm Aut}(S)$. Hence, $\varphi|_{S}$ is $q$-central. By Lemma 2.1, we have $\varphi|_{S}\in {\rm Inn}(S)$, that is, there exists a $h\in S$ with $\varphi|_{S} = {\rm conj}(h)|_{S}$. Again by Lemma 2.5, we may suppose that $\varphi|_{S} = id|_{S}$. By Lemma 2.2 and $S$ identifies with ${\rm Inn}(S)$, we obtain that $C_{A}(S) = C_{{\rm Aut}(S)}(S)\cap A = 1$. Thus, for any $y\in A$ and $x\in S$, we have $(y^{-1}xy)^{\rho} = (y^{-1})^{\varphi}xy^{\varphi} = y^{-1}xy$, which implies that $y^{\varphi}y^{-1}\in C_{A}(S) = 1$. Hence,

By Lemma 2.6(2), $\varphi|_{F/A}\in {\rm Aut_{Col}}(F/A)$. Note that $F/A$ is abelian, which implies that

Now, by Lemma 2.8, we obtain that

By Lemma 2.4, we have $O_{p}({\rm \zeta}(A)) = 1$. Hence, by (3.3), $\varphi = id$.

Corollary 3.2. Let $S$ be a simple normal subgroup of $F$. If $F/S$ is an abelian group, then ${\rm Out_{Col}}(F) = 1$. In particular, ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$.

Proof. If $S$ is abelian simple, this is a consequence of Proposition 3.1 in ^[6]. Next, we suppose that $S$ is non-abelian simple. Hence the assertion holds by Theorem 3.1.

Theorem 3.3. Let $A$ be an almost simple normal subgroup of $F$. If $F/A$ is a simple group, then ${\rm Out_{Col}}(F) = 1$. In particular, ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$.

Proof. Let $\rho\in {\rm Aut_{Col}}(F)$ and let $\rho$ be of $p$-power order, where $p\in \pi(F)$. We shall prove that $\rho\in {\rm Inn}(F)$. If $F/A$ is abelian simple, then the conclusion holds by Theorem 3.1. Next, we suppose that $F/A$ is non-abelian simple. It follows from Lemma 2.6(2) and Lemma 2.1 that $\rho|_{F/A} \in {\rm Inn}(F/A)$, that is, $\rho|_{F/A} = {\rm conj}(x)|_{F/A}$ for some $x\in F$. In general, by Lemma 2.9, we may suppose that

First, we show that $\rho|_{A}\in {\rm Aut_{Col}}(A)$. Since $\rho\in {\rm Aut_{Col}}(F)$, then there is a $k\in F$ such that

where $Q\in {\rm Syl}(A)$. Set $H = AC_{F}(Q)$. By Lemma 2.10,

Note that $H\geq A$. By (3.4), we deduce that

Consequently, by (3.6) and (3.7), we obtain that ${\rm conj}(k)|_{F/H} = id|_{F/H}$, this implies that $kH \in {\rm \zeta}(F/H)$. Note that $H/A\unlhd F/A$ and $F/A$ is non-abelian simple, then $H/A = 1$ or $H/A = F/A$. From this, we deduce that ${\rm \zeta}(F/H) = 1$. Hence, $k\in H$. Note further that $H = AC_{F}(Q) = C_{F}(Q)A$, we may suppose that $k = ra$, where $r\in C_{F}(Q)$, $a\in A$. By (3.5),

By (3.8), we have $\rho|_{A} \in {\rm Aut_{Col}}(A)$. Since $A$ is almost simple, then $\rho|_{A} \in {\rm Inn}(A)$ by Lemma 2.7, i.e., there is a $b\in A$ with $\rho|_{A} = {\rm conj}(b)|_{A}$. Set $\varphi = \rho{\rm conj}(b^{-1})$. In general, we suppose that $\varphi$ is of $p$-power order, and

By (3.4), we also have

Hence, by Lemma 2.8,

By Lemma 2.4, $O_{p}({\rm \zeta}(A)) = 1$. Thus, by (3.11), we have that $\varphi = id$, i.e., $\rho\in {\rm Inn}(F)$.

Corollary 3.4. Let $S$ be a simple normal subgroup of $F$. If $F/S$ is a simple group, then ${\rm Out_{Col}}(F) = 1$. In particular, ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$.

Proof. If $S$ is abelian simple, this is a consequence of Theorem 1.2 in ^[6]. Next, we suppose that $S$ is non-abelian simple. Consequently, the assertion holds by Theorem 3.3.

Theorem 3.5. Let $M$ be a normal subgroup of odd order of $F$. If $F/M$ is an almost simple group, then ${\rm Out_{Col}}(F)$ is of odd order. In particular, ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$.

Proof. Let $\rho\in {\rm Aut_{Col}}(F)$ and let $\rho$ be of $2$-power order. We shall prove that $\rho\in {\rm Inn}(F)$. By Lemma 2.6(2), $\rho|_{F/M}\in {\rm Aut_{Col}}(F/M)$. Since $F/M$ is almost simple, then, by Lemma 2.7, $\rho|_{F/M}\in {\rm Inn}(F/M)$, i.e., $\rho|_{F/M} = {\rm conj}(x)|_{F/M}$ for some $x\in F$. In general, we may suppose that

First, we show that $\rho|_{M}\in {\rm Aut_{Col}}(M)$. Since $\rho\in {\rm Aut_{Col}}(F)$, then

where $t\in F, P\in {\rm Syl}(M)$. Set $H = M{\rm C}_{F}(P)$, by Lemma 2.10, $H\unlhd F$ and $H^{\rho} = H$. Moreover,

Note that $H\geq M$. By (3.12), we have

By (3.14) and (3.15), ${\rm conj}(t)|_{G/H} = id|_{F/H}$, which implies that $tH\in {\rm \zeta}(F/H)$. Since $F/M$ is almost simple, then we may suppose that $S/M\leq F/M \leq {\rm Aut}(S/M)$. Note that $H/M\lhd F/M$ and $S/M\lhd F/M$, so either $H/M\cap S/M = 1$ or $S/M\leq H/M$. If $H/M\cap S/M = 1$, then $[H/M, S/M] = 1$. It follows from Lemma 2.2 that $H = M$. If $S/M\leq H/M$, then ${\rm \zeta}(H/M) = 1, {\rm \zeta}(F/M) = 1$ by Lemma 2.3. From this, we deduce that ${\rm \zeta}(F/H) = 1$, that is, $t\in H$. Note further that $H = M{\rm C}_{F}(P) = {\rm C}_{F}(P)M$, we may suppose that $t = cm$, where $c\in {\rm C}_{F}(P), m\in M$. By (3.13), we have

Thus (3.16) implies that $\rho|_{M}\in {\rm Aut_{Col}}(M)$. Next, by Lemma 2.11,

Hence, by Lemma 2.8,

But note that ${\rm O}_{2}({\rm \zeta}(M)) = 1$, so (3.18) implies that $\rho = id$.

Corollary 3.6. Let $M$ be a normal subgroup of odd order of $F$. If $F/M$ is a non-abelian simple group, then ${\rm Out_{Col}}(F)$ is of odd order. In particular, ${\rm Aut}_{\mathbb{Z}}(F) = {\rm Inn}(F)$.

Acknowledgments

Supported by NSF of China(11871292).

Conflict of interest

The authors declare there is no conflicts of interest.