# Finite groups with only $𝔉${\mathfrak{F}}-normal and $𝔉${\mathfrak{F}}-abnormal subgroups

Bin Hu 1 , Jianhong Huang 2 ,  and Alexander N. Skiba 3
• 1 School of Mathematics and Statistics, Jiangsu Normal University, 221116, Xuzhou, P. R. China
• 2 School of Mathematics and Statistics, Jiangsu Normal University, 221116, Xuzhou, P. R. China
• 3 Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University, 246019, Gomel, Belarus
Bin Hu
• Corresponding author
• School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, P. R. China
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, Jianhong Huang
and Alexander N. Skiba
• Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University, Gomel, 246019, Belarus
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## Abstract

Let G be a finite group, and let $𝔉$ be a class of groups. A chief factor $H/K$ of G is said to be $𝔉$-central (in G) if the semidirect product $(H/K)⋊(G/CG⁢(H/K))∈𝔉$. We say that a subgroup A of G is $𝔉$-normal in G if every chief factor $H/K$ of G between $AG$ and $AG$ is $𝔉$-central in G and $𝔉$-abnormal in G if V is not $𝔉$-normal in W for every two subgroups $V of G such that $A≤V$. We give a description of finite groups in which every subgroup is either $𝔉$-normal or $𝔉$-abnormal.

## 1 Introduction

Throughout this paper, all groups are finite, and G always denotes a finite group; $ℒ⁢(G)$ denotes the lattice of all subgroups of G. The group G is said to be strongly supersoluble [6] if G is supersoluble and G induces on any of its chief factor $H/K$ an automorphism group of square free order.

In what follows, $𝔉$ is a class of groups containing all nilpotent groups; $G𝔉$ denotes the intersection of all normal subgroups N of G with $G/N∈𝔉$. The class $𝔉$ is said to be a formation if every homomorphic image of $G/G𝔉$ belongs to $𝔉$ for every group G. The formation $𝔉$ is said to be saturated if $G∈𝔉$ whenever $G𝔉≤Φ⁢(G)$ and hereditary (Mal’cev [9]) if $H∈𝔉$ whenever $H≤G∈𝔉$.

Recall that $𝔑$, $𝔘$ and $𝔘s$ are the classes of all nilpotent, all supersoluble and all strongly supersoluble groups, respectively. It is well known that $𝔑$ and $𝔘$ are hereditary saturated formations. In the paper [13], it is proved that the class $𝔘s$ is also a hereditary saturated formation.

If $K≤H$ are normal subgroups of G and $C≤CG⁢(H/K)$, then we can form the semidirect product $(H/K)⋊(G/C)$ putting

$(h⁢K)g⁢C=g-1⁢h⁢g⁢K for all⁢h⁢K∈H/K⁢and⁢g⁢C∈G/C.$

We say that a chief factor $H/K$ of G is $𝔉$-central in G (see [12]) if

$(H/K)⋊(G/CG⁢(H/K))∈𝔉.$

Definition 1.1.

We say that a subgroup A of G is

1. (i)$𝔉⁢p$-normal in G if every chief factor $H/K$ of G between $AG$ and $AG$ with $p∈π⁢(H/K)$ is $𝔉$-central in G,
2. (ii)$𝔉$-normal in G provided A is $𝔉⁢p$-normal in G for all primes p,
3. (iii)$𝔉$-abnormal in G if V is not $𝔉$-normal in W for every two subgroups $V of G such that $A≤V$.

By definition, all normal subgroups of G are $𝔉$-normal in G. Moreover, G is the unique subgroup of G which simultaneously is $𝔉$-normal and $𝔉$-abnormal in G. Every maximal subgroup of G is either $𝔉$-normal or $𝔉$-abnormal in G. In this paper, we study those groups in which each subgroup is either $𝔉$-normal or $𝔉$-abnormal.

Example 1.2.

Before continuing, consider some well-known examples.

1. (i)A subgroup A of G is said to be quasinormal or permutable if A permutes with all subgroups H of G, that is, $A⁢H=H⁢A$. In view of [8] (see also [1, Corollary 1.5.6] or [10, Theorem 5.2.3]), every quasinormal subgroup of G is $𝔑$-normal in G.
2. (ii)A subgroup M of G is called modular if M is a modular element (in the sense of Kurosh [10, p. 43]) of the lattice $ℒ⁢(G)$ of all subgroups of G, that is,
1. (a)$〈X,M∩Z〉=〈X,M〉∩Z$ for all $X≤G$, $Z≤G$ such that $X≤Z$,
2. (b)$〈M,Y∩Z〉=〈M,Y〉∩Z$ for all $Y≤G$, $Z≤G$ such that $M≤Z$.
In view of [10, Theorem 5.1.14], every modular subgroup of G is $𝔘s$-normal in G. We say that a subgroup A of G is abmodular in G if V is not modular in W for every two subgroups $V of G such that $A≤V$. In view of [10, Lemma 5.1.2], a subgroup A of G is abmodular in G if and only if it is $𝔘s$-abnormal in G.
3. (iii)Let $G=(C7⋊(C2×C3))×P$, where $C2×C3=Aut⁡(C7)$ and P is a non-abelian group of order $p3$ of exponent p for some prime $p>2$. Then the subgroup $C2$ is not $𝔘s$-normal in G. Now let L be a subgroup of P of order p with $L≰Z⁢(P)$. Then L is neither quasinormal nor modular in G, but L evidently is $𝔑$-normal and so $𝔘s$-normal in G.

Definition 1.3.

We say that G is a DM-group if $G=D⋊M$, where

1. (1)$D=G′≠1$ is abelian,
2. (2)$M=〈x〉$ is a cyclic abnormal Sylow p-subgroup of G, where p is the smallest prime dividing $|G|$,
3. (3)$MG=〈xp〉=Z⁢(G)$,
4. (4)The element x induces a fixed-point-free power automorphism on D.

In [4], Fattahi defined B-groups to be groups in which every subgroup is either normal or abnormal, and he showed that a non-nilpotent group G is a B-group if and only if G is a DM-group. Later the results in [4] were generalized in several directions. In particular, Qinhai Zhang proved in [14] that every non-nilpotent group in which each subgroup is either quasinormal or abnormal is a B-group, and he posed a general problem of finding other conditions under which G is a B-group.

In this paper, we prove the following result in this research line.

Theorem 1.4.

Let $F$ be a hereditary saturated formation containing all nilpotent groups. If every subgroup of G is either $F$-normal or $F$-abnormal in G, then G is of either of the following types:

1. (I)$G∈𝔉$.
2. (II)$G=D⋊M$ is a DM-group, where $D=G𝔉$, and M is an $𝔉$-abnormal subgroup of G with $MG=Z𝔉⁢(G)$.

Conversely, in a group G of type (I) or (II), every subgroup is either $F$-normal or $F$-abnormal.

In this theorem, $Z𝔉⁢(G)$ denotes the $𝔉$-hypercenter of G, that is, the product of all normal subgroups N of G such that either $N=1$ or every chief factor of G below N is $𝔉$-central in G.

In view of [11, Chapter IV, Theorem 17.1], A is an $𝔑$-abnormal subgroup of a soluble group G if and only if A is abnormal in G. Therefore, we get from Theorem 1.4 the following:

Corollary 1.5.

Every subgroup of G is either $N$-normal or $N$-abnormal in G if and only if G is either nilpotent or a B-group.

Corollary 1.5 covers the main result in [4]. Moreover, in view of Example 1.2 (i), we get from Corollary 1.5 the following result.

Corollary 1.6 (Zhang [14]).

Let G be a non-nilpotent group. Then the following statements are equivalent:

1. (1)G is a B-group.
2. (2)Every subgroup of G is either quasinormal or abnormal in G.

Strongly supersoluble groups have found applications in many works (see, for example, [6, 13, 15]). Since every DM-group is evidently strongly supersoluble and the class of all strongly supersoluble groups is a hereditary saturated formation, we get from Theorem 1.4 and Proposition 3.4 the following characterizations of such groups.

Corollary 1.7.

The following statements are equivalent:

1. (i)G is strongly supersoluble.
2. (ii)Every Sylow subgroup of G is $𝔘s$-normal in G.
3. (iii)Each subgroup of G is either $𝔘s$-normal or $𝔘s$-abnormal in G.

In fact, G is an M-group [10], that is, the lattice $ℒ⁢(G)$ is modular if and only if every subgroup of G is modular in G. From Corollary 1.7 and Example 1.2 (ii), we get that G is an M-group also in the case when every non-abmodular subgroup of G is modular in G.

In conclusion of this section, note that one of the main tools in the proof of Theorem 1.4 is the following useful fact.

Proposition 1.8.

The class of all $F$-normal subgroups and, for any prime p, the class of all $F⁢p$-normal subgroups of G are sublattices of the lattice $L⁢(G)$.

## 2 Proof of Proposition 1.8

The first two lemmas can be proved by direct checking.

Lemma 2.1.

Let N, M and $K be normal subgroups of G, where $H/K$ is a chief factor of G.

1. (1)If $N≤K$, then
$(H/K)⋊(G/CG⁢(H/K))≃((H/N)/(K/N))⋊((G/N)/CG/N⁢((H/N)/(K/N))).$
2. (2)If $T/L$ is a chief factor of G and $H/K$ and $T/L$ are G-isomorphic, then $CG⁢(H/K)=CG⁢(T/L)$ and
$(H/K)⋊(G/CG⁢(H/K))≃(T/L)⋊(G/CG⁢(T/L)).$
3. (3)$(M⁢N/N)⋊(G/CG⁢(M⁢N/N))≃(M/M∩N)⋊(G/CG⁢(M/M∩N))$.

Recall that G is called a pd-group if $p∈π⁢(G)$.

Lemma 2.2.

Let $K≤H$, $K≤V$, $W≤V$ and $N≤H$ be normal subgroups of G. Suppose that every chief pd-factor of G between K and H is $F$-central in G.

1. (1)If every chief pd-factor of G between K and KN is $𝔉$-central in G, then every chief pd-factor of G between $K∩N$ and N is $𝔉$-central in G.
2. (2)If every chief pd-factor of G between W and V is $𝔉$-central in G, then every chief pd-factor of G between $K∩W$ and $H∩V$ is $𝔉$-central in G.
3. (3)If every chief pd-factor of G between K and V is $𝔉$-central in G, then every chief pd-factor of G between K and HV is $𝔉$-central in G.

Proof of Proposition 1.8.

Let $A,B$ be subgroups of G such that $A,B∈ℒp⁢c⁢𝔉⁢(G)$, where $ℒp⁢c⁢𝔉⁢(G)$ is the class of all $𝔉⁢p$-normal subgroups of G. Then every chief pd-factor of G between $AG$ and $AG$ is $𝔉$-central in G.

First we show that $A∩B∈ℒp⁢c⁢𝔉⁢(G)$. Note that $(A∩B)G=AG∩BG$. On the other hand, from the G-isomorphism

$(AG∩BG)/(AG∩BG)=(AG∩BG)/(AG∩BG∩AG)≃AG⁢(BG∩AG)/AG≤AG/AG,$

we get that every chief pd-factor of G between $AG∩BG$ and $AG∩BG$ is $𝔉$-central in G by Lemma 2.2 (1). Similarly, we get that every chief pd-factor of G between $BG∩AG$ and $BG∩AG$ is $𝔉$-central in G. But then we get that every chief pd-factor of G between $(AG∩BG)∩(BG∩AG)=AG∩BG$ and $AG∩BG$ is $𝔉$-central in G by Lemma 2.2 (2). It is clear also that $(A∩B)G≤AG∩BG$. Thus every chief pd-factor of G between $(A∩B)G=AG∩BG$ and $(A∩B)G$ is $𝔉$-central in G. Therefore, $A∩B∈ℒp⁢c⁢𝔉⁢(G)$.

Now we show that $〈A,B〉∈ℒp⁢c⁢𝔉⁢(G)$. In view of the G-isomorphisms

$AG⁢(AG⁢BG)/AG⁢BG≃AG/(AG∩AG⁢BG)=AG/AG⁢(AG∩BG)≃(AG/AG)/(AG⁢(AG∩BG)/AG),$

we get that every chief pd-factor of G between $AG⁢BG$ and $AG⁢(AG⁢BG)$ is $𝔉$-central in G. Similarly, every chief pd-factor of G between $AG⁢BG$ and $BG⁢(AG⁢BG)$ is $𝔉$-central in G. Moreover,

$AG⁢BG/AG⁢BG=(AG⁢(AG⁢BG)/AG⁢BG)⁢(BG⁢(AG⁢BG)/AG⁢BG),$

and so every chief pd-factor of G between $AG⁢BG$ and $AG⁢BG$ is $𝔉$-central in G by Lemma 2.2 (3).

Next note that $〈A,B〉G=AG⁢BG$ and $AG⁢BG≤〈A,B〉G$. Therefore, every chief pd-factor of G between $〈A,B〉G$ and $〈A,B〉G=AG⁢BG$ is $𝔉$-central in G. Hence $〈A,B〉∈ℒp⁢c⁢𝔉⁢(G)$.

Therefore, $ℒp⁢c⁢𝔉⁢(G)$ is a sublattice of the lattice $ℒ⁢(G)$. Finally, for the class $ℒc⁢𝔉⁢(G)$, of all $𝔉$-normal subgroups of G, we have

$ℒc⁢𝔉⁢(G)=⋂p∈π⁢(G)ℒp⁢c⁢𝔉⁢(G),$

and so $ℒc⁢𝔉⁢(G)$ is also a sublattice of $ℒ⁢(G)$. ∎

## 3 Proof of Theorem 1.4

The following lemma is well known (see, for example, [12, Lemma 3.29]).

Lemma 3.1.

Let $H/K$ be an abelian chief factor of G, and let V be a maximal subgroup of G with $K≤M$ and $H⁢M=G$. Then

$G/VG≃(H/K)⋊(G/CG⁢(H/K)).$

Lemma 3.2 ([5, Chapter 1, Theorem 2.7]).

Let $F$ be a hereditary saturated formation, and let $Z=ZF⁢(G)$. Let N and E be subgroups of G, where N is normal in G.

1. (1)If $N≤Z$, then $Z/N=Z𝔉⁢(G/N)$.
2. (2)$Z∩E≤Z𝔉⁢(E)$.
3. (3)If $N⁢Z/N≤Z𝔉⁢(G/N)$.

In fact, the following lemma is a corollary of [3, Chapter IV, Theorem 6.7].

Lemma 3.3.

Let $F$ be a hereditary saturated formation, and let A and $N≤E$ be subgroups of G, where N is normal and A is $F$-normal in G. Then

1. (1)$A⁢N/N$ is $𝔉$-normal in $G/N$.
2. (2)If $E/N$ is $𝔉$-normal in $G/N$, then E is $𝔉$-normal in G.
3. (3)$A∩E$ is $𝔉$-normal in E.

Proof.

(1) From the G-isomorphisms

$(AG⁢N/N)/(AG⁢N/N)≃AG⁢N/AG⁢N≃AG/(AG∩AG⁢N)=AG/AG⁢(AG∩N)≃(AG/AG)/(AG⁢(AG∩N)/AG)$

and Lemma 2.1, we get that every chief factor of $G/N$ between $AG⁢N/N$ and $AG⁢N/N$ is $𝔉$-central in $G/N$ since every chief factor of G between $AG$ and $AG$ is $𝔉$-central in $G/N$ by hypothesis. On the other hand, we have

$(A⁢N/N)G/N=(A⁢N)G/N=AG⁢N/N and AG⁢N/N≤(A⁢N/N)G/N.$

Hence every chief factor of $G/N$ between $(A⁢N/N)G/N$ and $(A⁢N/N)G/N$ is $𝔉$-central in $G/N$, so $A⁢N/N$ is $𝔉$-normal in $G/N$.

(2) This follows from the G-isomorphism

$EG/EG≃(EG/N)/(EG/N)=(E/N)G/N/(E/N)G/N.$

(3) First note that

$(AG/AG)∩(E⁢AG/AG)=AG⁢(AG∩E)/AG≤Z𝔉⁢(E⁢AG/AG)$

by Lemma 3.2 (2) since, by hypothesis, we have $AG/AG≤Z𝔉⁢(G/AG)$. On the other hand, we have

$f⁢(Z𝔉⁢(E⁢AG/AG))=Z𝔉⁢(E/AG∩E),$

where $f:E⁢AG/AG→E/E∩AG$ is the canonical isomorphism from $E⁢AG/AG$ onto $E/E∩AG$. Hence

$f⁢(AG⁢(AG∩E)/AG)=(AG∩E)/(AG∩E)≤Z𝔉⁢(E/AG∩E),$

where

$AG∩E≤(E∩A)E≤A∩E≤(A∩E)E≤AG∩E,$

and so

$(A∩E)E/(A∩E)E≤Z𝔉⁢(E/(A∩E)E)$

by Lemma 3.2 (2) and (3). Hence $A∩E$ is $𝔉$-normal in E. ∎

Proposition 3.4.

Let $F$ be a saturated formation containing all nilpotent groups. Then the following statements are equivalent:

1. (i)$G∈𝔉$.
2. (ii)Every chief factor of G is $𝔉$-central in G.
3. (iii)Every Sylow subgroup of G is $𝔉$-normal in G.

Proof.

(i) $⇒$ (ii) This directly follows from the Barnes–Kegel result [3, Chapter IV, Proposition 1.5].

(ii) $⇒$ (iii) This implication is evident.

(ii) $⇒$ (i) In fact, this application is well known, and it can be easily proved by using Lemma 3.1 and induction on $|G|$.

(iii) $⇒$ (i) Let P be a Sylow p-subgroup of G, where p is any prime dividing $|G|$. Then $P⁢R/R$ is a Sylow p-subgroup of $G/R$ and $P⁢R/R$ is $𝔉$-normal in $G/R$ by Lemma 3.3 (1) since P is $𝔉$-normal in G by hypothesis. Therefore, the hypothesis holds for $G/R$, so $G/R∈𝔉$ by induction. Therefore, if either $R≤Φ⁢(G)$ or G has a minimal normal subgroup $N≠R$, then $G∈𝔉$. Moreover, if $R/1$ is $𝔉$-central in G, then, by the Jordan–Hölder theorem for the chief series, every chief factor of G is $𝔉$-central in G, and so $G∈𝔉$ by the implication (ii) $⇒$ (i).

Now assume that $R≰Φ⁢(G)$ is the unique minimal normal subgroup of G and that $R/1$ is not $𝔉$-central in G. Let $p∈π⁢(R)$, and let $Gp$ be a Sylow p-subgroup of G. Then $Rp=R∩Gp$ is normal in $Gp$. Suppose that $Rp≠R$; then, for some maximal subgroup V of G, we have $Gp≤NG⁢(Rp)≤V$, and so $G=R⁢V$ by the Frattini argument. Then $(Gp)G≤VG=1$, so $1. But then $R≤Z𝔉⁢(G)$, and so $R/1$ is $𝔉$-central in G, a contradiction. Hence $R=Rp$ is an abelian p-group for some prime p. Let M be a maximal subgroup of G such that $G=R⁢M=R⋊M$. If G is a p-group, then $G∈𝔉$ since $𝔉$ contains all nilpotent groups by hypothesis. Now assume $|π⁢(G)|>1$, and let Q be a Sylow q-subgroup of $M$, where $q∈π⁢(G)∖{p}$. Then $QG=1$, and so $1, which again implies that $R/1$ is $𝔉$-central in G, a contradiction. The implication is proved. ∎

In fact, the following lemma is a corollary of [3, Chapter IV, Theorem 6.7].

Lemma 3.5.

Let P be a normal p-subgroup of G. If every chief factor of G between $Φ⁢(P)$ and P is cyclic, then every chief factor of G below P is cyclic.

Proof of Theorem 1.4.

Necessity. Assume that this is false, and let G be a counterexample of minimal order. Then $G∉𝔉$, so $D=G𝔉≠1$, and also G is not nilpotent since $𝔉$ contains all nilpotent groups by hypothesis. Let R be a minimal normal subgroup of G.

(1) Every proper subgroup $E∉𝔉$ is of type (II) and if $G/R∉𝔉$, then $G/R$ is of type (II): Let A be any subgroup of E. If A is $𝔉$-abnormal in G, then A is evidently $𝔉$-abnormal in E. On the other hand, if A is $𝔉$-normal in G, then A is $𝔉$-normal in E by Lemma 3.3 (3). Hence the hypothesis holds for E, so the choice of G implies that E is of type (II).

Finally, if $A/R$ is a non-$𝔉$-abnormal subgroup of $G/R$, then A is not $𝔉$-abnormal in G, and so, by hypothesis and Lemma 3.3 (1), $A/R$ is $𝔉$-normal in $G/R$. Therefore, $G/R$ is of type (II) by the choice of G.

(2) $D: Assume $D=G$, and let P be a Sylow p-subgroup of G, where p is the smallest prime dividing $|G|$. Then P is not cyclic since otherwise G has a normal p-complement E by [7, Chapter IV, Satz 2.8] and $G/E∈𝔉$, which implies $D≤E. Then, for a maximal subgroup V of P, we have $V≠1$, and V is not $𝔉$-abnormal in G, so V is $𝔉$-normal in G. Assume $VG≠1$ and $R≤VG$. If $G/R∈𝔉$, then $D=R, which is impossible by our assumption $D=G$. Hence $G/R∉𝔉$, so statement (II) holds for $G/R$ by claim (1), which implies $G/R=D/R=(G/R)𝔉=(G/R)′ by [2, Proposition 2.2.8]. Then $D=G. This contradiction shows that $VG=1$, so $1, and hence we can assume without loss of generality $R≤Z𝔉⁢(G)$. Then $G/R∉𝔉$ by Lemma 3.2 (1), and so, as above, we get $D=G. This contradiction completes the proof of Claim (2).

(3) D is nilpotent, and every element of G induces a power automorphism in $D/Φ⁢(D)$. Moreover, if p is the smallest prime dividing $|G|$, then p does not divide $|D|$: Let V be a maximal subgroup of D. Then V is not $𝔉$-abnormal in G by claim (2), so V is $𝔉$-normal in G. Assume that V is not normal in G. Then we have $VG=D$ and $VG/VG≤Z𝔉⁢(G/VG)$, which implies $G/VG∈𝔉$ by Lemma 3.2 (1). But then $D≤VG≤D$. This contradiction shows that V is normal in G, so D is nilpotent, and every element of G induces a power automorphism in $D/Φ⁢(D)$ since every subgroup of $D/Φ⁢(D)$ can be written as the intersection of some maximal subgroup of $D/Φ⁢(D)$.

Now assume that p divides $|D|$. Then p divides $|D/Φ⁢(D)|$, so, from the previous paragraph, we know that, for some maximal subgroup V of D, we have $|D:V|=p$, and V is normal in G. But then $G/CG⁢(D/V)$ is a cyclic group of order dividing $p-1$. Since p is the smallest prime dividing $|G|$, it follows that $CG⁢(D/V)=G$, and so $D/V≤Z⁢(G/V)$. Hence $G/V∈𝔉$ by Lemma 3.2 (1) since $𝔉$ contains all nilpotent groups by hypothesis, and so $D≤V, a contradiction. Thus $|D|$ is a $p′$-number. Hence we have (3).

(4) D is a Hall subgroup of G. Hence D has a complement M in G, and p divides $|M|$: Suppose that this is false, and let P be a Sylow r-subgroup of D such that $1, where $Gr∈Sylr⁡(G)$.

First we show that D is a minimal normal subgroup of G. Assume that this is false. Then, for a minimal normal subgroup N of G contained in D, we have $G/N∉𝔉$. Since D is nilpotent by claim (3), N is a q-group for some prime q. Moreover, $D/N=(G/N)𝔉$ is a Hall subgroup of $G/N$ by claim (1) and [2, Proposition 2.2.8]. Suppose that $P⁢N/N≠1$. Then we have $P⁢N/N∈Sylr⁡(G/N)$. If $q≠r$, then $P∈Sylr⁡(G)$. This contradicts the fact that $P. Hence $q=r$, so $N≤P$, and therefore $P/N∈Sylr⁡(G/N)$. It follows that $P∈Sylr⁡(G)$. This contradiction shows that $P⁢N/N=1$, which implies that $N=P$ is a Sylow r-subgroup of D.

Therefore, N is the unique minimal normal subgroup of G contained in D. It is also clear that a p-complement U of D is a Hall subgroup of G. Claim (3) implies that U is characteristic in D, so it is normal in G. Therefore, $U=1$, and hence $D=N$. So $N≰Φ⁢(G)$ since the formation $𝔉$ is saturated by hypothesis.

Let V be a maximal subgroup of N. Then V is $𝔉$-normal in G by claim (2). Assume $V≠1$. Then $VG, where $N/1=VG/VG≤Z𝔉⁢(G/1)$, which implies $G∈𝔉$ by Lemma 3.2 (1). This contradiction shows that $V=1$, so we have $|N|=r$.

Now let S be a maximal subgroup of G such that $N≰S$, and let $C=CG⁢(N)$. Then $G=N⋊S$, and so $C=N⁢(C∩S)$, where $C∩S$ is normal in G. Assume $C∩S≠1$, and let L be a minimal normal subgroup of G contained in $C∩S$. Then $N⁢L/L=D⁢L/L=(G/L)𝔉$ is a Hall subgroup of $G/L$, and so $N⁢L/L=Gr⁢L/L$, which implies $N⁢L=Gr⁢L$. Hence $Gr=N×(Gr∩L)$. Let $N=〈a〉$, and let $A=〈b〉$ be a subgroup of order r of $Gr∩L$. Let $L0=〈a⁢b〉$. Then $|L0|=r$, and $L0∩N=1=L0∩Gr∩L$. First assume that $L0$ is not normal in G. Then $L0≤L0G≤Z𝔉⁢(G)$. On the other hand, from the G-isomorphism $N⁢L/N≃L$, it follows that $L≤Z𝔉⁢(G)$ by Lemma 2.1 and Proposition 3.4. Moreover, $Gr=L0⁢(Gr∩L)$ since $|N|=r$ and $L0∩(Gr∩L)=1$. Hence we have $N≤Gr≤Z𝔉⁢(G)$, and so $G∈𝔉$ by Lemma 3.2 (1). This contradiction shows that $C∩S=1$, and so $C=N$ is the unique minimal normal subgroup of G. Since $|N|=r$, it follows that $G/N=G/C$ is cyclic, so G is supersoluble. Then a Sylow q-subgroup Q of G, where q is the largest prime dividing $|G|$, is normal in G. Therefore, $N≤Q$ and $N≰Φ⁢(Q)≤Φ⁢(G)$. Hence $q=r$, and Q is an elementary abelian r-group, which implies $D=N=Q=Gr$. This contradiction completes the proof of the fact that D is a Hall subgroup of G. Therefore, D has a complement M in G by the Schur–Zassenhaus theorem. Moreover, p divides $|M|$ by claim (3). Hence we have (4).

(5) $M=〈x〉$ is a cyclic $𝔉$-abnormal Sylow p-subgroup of G, and every proper subgroup of M is $𝔉$-normal in G: By claim (4), $M≃G/D∈𝔉$, and p divides $|M|$. Therefore, every proper subgroup V of M is not $𝔉$-abnormal in G, so V is $𝔉$-normal in G by hypothesis. Let $Mp$ be a Sylow p-subgroup of M. Assume $Mp≠M$. Then $Mp$ is $𝔉$-normal in G. Moreover, in this case, each Sylow subgroup P of M is $𝔉$-normal in G, and P is a Sylow subgroup of G by claim (4). It follows that all Sylow subgroups of G are $𝔉$-normal in G by claims (3) and (4), and therefore $G∈𝔉$ by Proposition 3.4. This contradiction shows that $M=Mp$ is $𝔉$-abnormal in G and every proper subgroup of M is $𝔉$-normal in G. Therefore, M is cyclic since the set of all $𝔉$-normal subgroups of G forms a sublattice of the lattice of subgroups of G by Proposition 1.8. Hence we have (5).

(6) $Z𝔉⁢(G)=MG=〈xp〉$: Let $Z=Z𝔉⁢(G)$, $V=〈xp〉$. Assume $Z∩D≠1$. Let $U=(Z∩D)⁢M$. Then $Z∩D≤Z∩U≤Z𝔉⁢(U)$ by Lemma 3.2 (2), so $U/Z𝔉⁢(U)≃M/M∩Z𝔉⁢(U)∈𝔉$, and hence $U∈𝔉$ by Lemma 3.2 (1). But then M is not $𝔉$-abnormal in G, contrary to claim (5). Therefore, $Z∩D=1$, so $Z≤M$, and hence, in fact, $Z≤VG$. It is clear that $D≤CG⁢(VG)$, so $VG≤Z⁢(G)$ by claim (5). Therefore, $V≤VG≤Z≤V$ by Lemma 3.2 (1). Hence we have $MG=VG=V=Z⁢(G)=Z$.

(7) $My≤NG⁢(H)$ for every proper subgroup H of D and every $y∈G$: Let V be a maximal subgroup of D such that $H≤V$. Then V is normal in G by claim (3). Let $E=V⁢My$ and $D0=E𝔉$. It is clear that $D0≤V$. Moreover, $D0≠1$ since otherwise $E∈𝔉$, which implies that the subgroups $My$ and M are not $𝔉$-abnormal in G, contrary to claim (5). Furthermore, claim (1) implies $E=D0⋊My=V⋊My$, and every subgroup of $D0=V$ is normalized by $My$. Hence $My≤NG⁢(H)$.

(8) Every proper subgroup H of D is normal in G, and hence D is abelian: Claim (7) implies $MG≤NG⁢(H)$, where M is $𝔉$-abnormal in G by claim (5), and so $G=MG≤NG⁢(H)$. Therefore, D is a Dedekind group, and $|D|$ is odd by claim (3). Hence we have (8).

From claims (4), (5), (6) and (8), it follows that condition (II) holds for G, which is impossible by the choice of G. This contradiction completes the proof of the necessity condition of the theorem.

Sufficiency. If $G∈𝔉$, then every subgroup of G is $𝔉$-normal in G by Proposition 3.4. Now assume that G is a group of type (II), and let A be any subgroup of G. First suppose that $|M|$ divides $|A|$. Then, for some $a∈G$, we have $Ma≤A$, so A is $𝔉$-abnormal in G since the subgroups M and $Ma$ are $𝔉$-abnormal in G. Now assume that $|M|$ does not divide $|A|$. Then, for a Sylow p-subgroup $Ap$ of A and for some $b∈G$, we have $(Ap)b, so the subgroups $(Ap)b$ and $Ap$ are $𝔉$-normal in G by hypothesis. Now note that $A=(A∩D)⁢Ap$, where $A∩D$ is normal in G, and so A is $𝔉$-normal in G by Proposition 1.8. ∎

Acknowledgements

The authors are very grateful to the helpful suggestions of the referee.

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B. Huppert, Endliche Gruppen. I, Springer, Berlin, 1967.

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R. Maier and P. Schmid, The embedding of permutable subgroups in finite groups, Math. Z. 131 (1973), 269–272.

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A. I. Mal’cev, Algebraic Systems, Nauka, Moscow, 1970.

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R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter, Berlin, 1994.

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Q. Zhang, Finite groups with only s-quasinormal and abnormal subgroups, Northeast. Math. J. 14 (1998), no. 1, 41–46.

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I. Zimmermann, Submodular subgroups in finite groups, Math. Z. 202 (1989), 545–557.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

• [1]

A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, Walter de Gruyter, Berlin, 2010.

• [2]

A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, Springer, Dordrecht, 2006.

• [3]

K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin, 1992.

• [4]

A. Fattahi, Groups with only normal and abnormal subgroups, J. Algebra 28 (1974), no. 1, 15–19.

• Crossref
• Export Citation
• [5]

W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg, 2015.

• [6]

B. Hu, J. Huang and A. N. Skiba, On generalized S-quasinormal and generalized subnormal subgroups of finite groups, Comm. Algebra 46 (2018), no. 4, 1758–1769.

• Crossref
• Export Citation
• [7]

B. Huppert, Endliche Gruppen. I, Springer, Berlin, 1967.

• [8]

R. Maier and P. Schmid, The embedding of permutable subgroups in finite groups, Math. Z. 131 (1973), 269–272.

• Crossref
• Export Citation
• [9]

A. I. Mal’cev, Algebraic Systems, Nauka, Moscow, 1970.

• [10]

R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter, Berlin, 1994.

• [11]

L. A. Shemetkov, Formations of Finite Groups, Moscow, Nauka, 1978.

• [12]

L. A. Shemetkov and A. N. Skiba, Formations of Algebraic Systems, Nauka, Moscow, 1989.

• [13]

V. A. Vasilyev, Finite groups with submodular Sylow subgroups, Siberian Math. J. 56 (2015), no. 6, 1019–1027.

• Crossref
• Export Citation
• [14]

Q. Zhang, Finite groups with only s-quasinormal and abnormal subgroups, Northeast. Math. J. 14 (1998), no. 1, 41–46.

• [15]

I. Zimmermann, Submodular subgroups in finite groups, Math. Z. 202 (1989), 545–557.

• Crossref
• Export Citation
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