# Finite groups with only π-normal and π-abnormal subgroups

Bin Hu , Jianhong Huang and Alexander N. Skiba
From the journal Journal of Group Theory

## Abstract

Let G be a finite group, and let π be a class of groups. A chief factor H/K of G is said to be π-central (in G) if the semidirect product (H/K)β(G/CGβ’(H/K))βπ. We say that a subgroup A of G is π-normal in G if every chief factor H/K of G between AG and AG is π-central in G and π-abnormal in G if V is not π-normal in W for every two subgroups V<W of G such that Aβ€V. We give a description of finite groups in which every subgroup is either π-normal or π-abnormal.

## 1 Introduction

Throughout this paper, all groups are finite, and G always denotes a finite group; ββ’(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, π is a class of groups containing all nilpotent groups; Gπ denotes the intersection of all normal subgroups N of G with G/Nβπ. The class π is said to be a formation if every homomorphic image of G/Gπ belongs to π for every group G. The formation π is said to be saturated if Gβπ whenever Gπβ€Ξ¦β’(G) and hereditary (Malβcev [9]) if Hβπ whenever Hβ€Gβπ.

Recall that π, π and πs are the classes of all nilpotent, all supersoluble and all strongly supersoluble groups, respectively. It is well known that π and π are hereditary saturated formations. In the paper [13], it is proved that the class πs is also a hereditary saturated formation.

If Kβ€H are normal subgroups of G and Cβ€CGβ’(H/K), then we can form the semidirect product (H/K)β(G/C) putting

(hβ’K)gβ’C=g-1β’hβ’gβ’Kβfor allβ’hβ’KβH/Kβ’andβ’gβ’CβG/C.

We say that a chief factor H/K of G is π-central in G (see [12]) if

(H/K)β(G/CGβ’(H/K))βπ.

## Definition 1.1.

We say that a subgroup A of G is

1. πβ’p-normal in G if every chief factor H/K of G between AG and AG with pβΟβ’(H/K) is π-central in G,

2. π-normal in G provided A is πβ’p-normal in G for all primes p,

3. π-abnormal in G if V is not π-normal in W for every two subgroups V<W of G such that Aβ€V.

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

## Example 1.2.

Before continuing, consider some well-known examples.

1. A subgroup A of G is said to be quasinormal or permutable if A permutes with all subgroups H of G, that is, Aβ’H=Hβ’A. In view of [8] (see also [1, Corollary 1.5.6] or [10, Theorem 5.2.3]), every quasinormal subgroup of G is π-normal in G.

2. A subgroup M of G is called modular if M is a modular element (in the sense of Kurosh [10, p.β43]) of the lattice ββ’(G) of all subgroups of G, that is,

In view of [10, Theorem 5.1.14], every modular subgroup of G is πs-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<W of G such that Aβ€V. In view of [10, Lemma 5.1.2], a subgroup A of G is abmodular in G if and only if it is πs-abnormal in G.

3. Let G=(C7β(C2ΓC3))ΓP, where C2ΓC3=Autβ‘(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 πs-normal in G. Now let L be a subgroup of P of order p with Lβ°Zβ’(P). Then L is neither quasinormal nor modular in G, but L evidently is π-normal and so πs-normal in G.

## Definition 1.3.

We say that G is a DM-group if G=DβM, where

1. D=Gβ²β 1 is abelian,

2. M=γxγ is a cyclic abnormal Sylow p-subgroup of G, where p is the smallest prime dividing |G|,

3. MG=γxpγ=Zβ’(G),

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. Gβπ.

2. G=DβM is a DM-group, where D=Gπ, and M is an π-abnormal subgroup of G with MG=Zπβ’(G).

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

In this theorem, Zπβ’(G) denotes the π-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 π-central in G.

In view of [11, Chapter IV, Theorem 17.1], A is an π-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 1.2β(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. G is a B-group.

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. G is strongly supersoluble.

2. Every Sylow subgroup of G is πs-normal in G.

3. Each subgroup of G is either πs-normal or πs-abnormal in G.

In fact, G is an M-group [10], that is, the lattice ββ’(G) is modular if and only if every subgroup of G is modular in G. From Corollary 1.7 and Example 1.2β(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 Fβ’p-normal subgroups of G are sublattices of the lattice Lβ’(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<Hβ€G be normal subgroups of G, where H/K is a chief factor of G.

1. If Nβ€K, then

(H/K)β(G/CGβ’(H/K))β((H/N)/(K/N))β((G/N)/CG/Nβ’((H/N)/(K/N))).
2. If T/L is a chief factor of G and H/K and T/L are G-isomorphic, then CGβ’(H/K)=CGβ’(T/L) and

(H/K)β(G/CGβ’(H/K))β(T/L)β(G/CGβ’(T/L)).

Recall that G is called a pd-group if pβΟβ’(G).

## Lemma 2.2.

Let Kβ€H, Kβ€V, Wβ€V and Nβ€H be normal subgroups of G. Suppose that every chief pd-factor of G between K and H is F-central in G.

1. If every chief pd-factor of G between K and KN is π-central in G, then every chief pd-factor of G between Kβ©N and N is π-central in G.

2. If every chief pd-factor of G between W and V is π-central in G, then every chief pd-factor of G between Kβ©W and Hβ©V is π-central in G.

3. If every chief pd-factor of G between K and V is π-central in G, then every chief pd-factor of G between K and HV is π-central in G.

## Proof of Proposition 1.8.

Let A,B be subgroups of G such that A,Bββpβ’cβ’πβ’(G), where βpβ’cβ’πβ’(G) is the class of all πβ’p-normal subgroups of G. Then every chief pd-factor of G between AG and AG is π-central in G.

Now we show that γA,Bγββpβ’cβ’πβ’(G). In view of the G-isomorphisms

we get that every chief pd-factor of G between AGβ’BG and AGβ’(AGβ’BG) is π-central in G. Similarly, every chief pd-factor of G between AGβ’BG and BGβ’(AGβ’BG) is π-central in G. Moreover,

AGβ’BG/AGβ’BG=(AGβ’(AGβ’BG)/AGβ’BG)β’(BGβ’(AGβ’BG)/AGβ’BG),

and so every chief pd-factor of G between AGβ’BG and AGβ’BG is π-central in G by Lemma 2.2β(3).

Next note that γA,BγG=AGβ’BG and AGβ’BGβ€γA,BγG. Therefore, every chief pd-factor of G between γA,BγG and γA,BγG=AGβ’BG is π-central in G. Hence γA,Bγββpβ’cβ’πβ’(G).

Therefore, βpβ’cβ’πβ’(G) is a sublattice of the lattice ββ’(G). Finally, for the class βcβ’πβ’(G), of all π-normal subgroups of G, we have

βcβ’πβ’(G)=βpβΟβ’(G)βpβ’cβ’πβ’(G),

and so βcβ’πβ’(G) is also a sublattice of ββ’(G). β

## 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 Kβ€M and Hβ’M=G. Then

G/VGβ(H/K)β(G/CGβ’(H/K)).

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

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

1. If Nβ€Z, then Z/N=Zπβ’(G/N).

3. If Nβ’Z/Nβ€Zπβ’(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 Nβ€E be subgroups of G, where N is normal and A is F-normal in G. Then

1. Aβ’N/N is π-normal in G/N.

2. If E/N is π-normal in G/N, then E is π-normal in G.

3. Aβ©E is π-normal in E.

## Proof.

(1) From the G-isomorphisms

and Lemma 2.1, we get that every chief factor of G/N between AGβ’N/N and AGβ’N/N is π-central in G/N since every chief factor of G between AG and AG is π-central in G/N by hypothesis. On the other hand, we have

(Aβ’N/N)G/N=(Aβ’N)G/N=AGβ’N/NβandβAGβ’N/Nβ€(Aβ’N/N)G/N.

Hence every chief factor of G/N between (Aβ’N/N)G/N and (Aβ’N/N)G/N is π-central in G/N, so Aβ’N/N is π-normal in G/N.

(2) This follows from the G-isomorphism

EG/EGβ(EG/N)/(EG/N)=(E/N)G/N/(E/N)G/N.

(3) First note that

by Lemma 3.2β(2) since, by hypothesis, we have AG/AGβ€Zπβ’(G/AG). On the other hand, we have

where

and so

by Lemma 3.2β(2) and (3). Hence Aβ©E is π-normal in E. β

## Proposition 3.4.

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

1. Gβπ.

2. Every chief factor of G is π-central in G.

3. Every Sylow subgroup of G is π-normal in G.

## Proof.

(i) β (ii)βThis directly follows from the BarnesβKegel result [3, Chapter IV, Proposition 1.5].

(ii) β (iii)βThis implication is evident.

(ii) β (i)βIn fact, this application is well known, and it can be easily proved by using Lemma 3.1 and induction on |G|.

(iii) β (i)βLet P be a Sylow p-subgroup of G, where p is any prime dividing |G|. Then Pβ’R/R is a Sylow p-subgroup of G/R and Pβ’R/R is π-normal in G/R by Lemma 3.3β(1) since P is π-normal in G by hypothesis. Therefore, the hypothesis holds for G/R, so G/Rβπ by induction. Therefore, if either Rβ€Ξ¦β’(G) or G has a minimal normal subgroup Nβ R, then Gβπ. Moreover, if R/1 is π-central in G, then, by the JordanβHΓΆlder theorem for the chief series, every chief factor of G is π-central in G, and so Gβπ by the implication (ii) β (i).

Now assume that Rβ°Ξ¦β’(G) is the unique minimal normal subgroup of G and that R/1 is not π-central in G. Let pβΟβ’(R), and let Gp be a Sylow p-subgroup of G. Then Rp=Rβ©Gp is normal in Gp. Suppose that Rpβ R; then, for some maximal subgroup V of G, we have Gpβ€NGβ’(Rp)β€V, and so G=Rβ’V by the Frattini argument. Then (Gp)Gβ€VG=1, so 1<Gpβ€Zπβ’(G). But then Rβ€Zπβ’(G), and so R/1 is π-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=Rβ’M=RβM. If G is a p-group, then Gβπ since π contains all nilpotent groups by hypothesis. Now assume |Οβ’(G)|>1, and let Q be a Sylow q-subgroup of M, where qβΟβ’(G)β{p}. Then QG=1, and so 1<QGβ€Zπβ’(G), which again implies that R/1 is π-central in G, a contradiction. The implication is proved. β

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 Ξ¦β’(P) and P is cyclic, then every chief factor of G below P is cyclic.

## Proof of Theorem 1.4.

Necessity.βAssume that this is false, and let G be a counterexample of minimal order. Then Gβπ, so D=Gπβ 1, and also G is not nilpotent since π contains all nilpotent groups by hypothesis. Let R be a minimal normal subgroup of G.

(1) Every proper subgroup Eβπ is of type (II) and if G/Rβπ, then G/R is of type (II):βLet A be any subgroup of E. If A is π-abnormal in G, then A is evidently π-abnormal in E. On the other hand, if A is π-normal in G, then A is π-normal in E by Lemma 3.3β(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-π-abnormal subgroup of G/R, then A is not π-abnormal in G, and so, by hypothesis and Lemma 3.3β(1), A/R is π-normal in G/R. Therefore, G/R is of type (II) by the choice of G.

(2) D<G:β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/Eβπ, which implies Dβ€E<G. Then, for a maximal subgroup V of P, we have Vβ 1, and V is not π-abnormal in G, so V is π-normal in G. Assume VGβ 1 and Rβ€VG. If G/Rβπ, then D=R<G, which is impossible by our assumption D=G. Hence G/Rβπ, so statement (II) holds for G/R by claim (1), which implies G/R=D/R=(G/R)π=(G/R)β²<G/R by [2, Proposition 2.2.8]. Then D=G<G. This contradiction shows that VG=1, so 1<VGβ€Zπβ’(G), and hence we can assume without loss of generality Rβ€Zπβ’(G). Then G/Rβπ by Lemma 3.2β(1), and so, as above, we get D=G<G. This contradiction completes the proof of Claim (2).

(3) D is nilpotent, and every element of G induces a power automorphism in D/Ξ¦β’(D). Moreover, if p is the smallest prime dividing |G|, then p does not divide |D|:βLet V be a maximal subgroup of D. Then V is not π-abnormal in G by claim (2), so V is π-normal in G. Assume that V is not normal in G. Then we have VG=D and VG/VGβ€Zπβ’(G/VG), which implies G/VGβπ by Lemma 3.2β(1). But then Dβ€VGβ€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/Ξ¦β’(D) since every subgroup of D/Ξ¦β’(D) can be written as the intersection of some maximal subgroup of D/Ξ¦β’(D).

Now assume that p divides |D|. Then p divides |D/Ξ¦β’(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/CGβ’(D/V) is a cyclic group of order dividing p-1. Since p is the smallest prime dividing |G|, it follows that CGβ’(D/V)=G, and so D/Vβ€Zβ’(G/V). Hence G/Vβπ by Lemma 3.2β(1) since π contains all nilpotent groups by hypothesis, and so Dβ€V<D, a contradiction. Thus |D| is a pβ²-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|:βSuppose that this is false, and let P be a Sylow r-subgroup of D such that 1<P<Gr, where GrβSylrβ‘(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/Nβπ. Since D is nilpotent by claim (3), N is a q-group for some prime q. Moreover, D/N=(G/N)π is a Hall subgroup of G/N by claim (1) and [2, Proposition 2.2.8]. Suppose that Pβ’N/Nβ 1. Then we have Pβ’N/NβSylrβ‘(G/N). If qβ r, then PβSylrβ‘(G). This contradicts the fact that P<Gr. Hence q=r, so Nβ€P, and therefore P/NβSylrβ‘(G/N). It follows that PβSylrβ‘(G). This contradiction shows that Pβ’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 Nβ°Ξ¦β’(G) since the formation π is saturated by hypothesis.

Let V be a maximal subgroup of N. Then V is π-normal in G by claim (2). Assume Vβ 1. Then VG<V<VG, where N/1=VG/VGβ€Zπβ’(G/1), which implies Gβπ by Lemma 3.2β(1). This contradiction shows that V=1, so we have |N|=r.

Now let S be a maximal subgroup of G such that Nβ°S, and let C=CGβ’(N). Then G=NβS, and so C=Nβ’(Cβ©S), where Cβ©S is normal in G. Assume Cβ©Sβ 1, and let L be a minimal normal subgroup of G contained in Cβ©S. Then Nβ’L/L=Dβ’L/L=(G/L)π is a Hall subgroup of G/L, and so Nβ’L/L=Grβ’L/L, which implies Nβ’L=Grβ’L. Hence Gr=NΓ(Grβ©L). Let N=γaγ, and let A=γbγ be a subgroup of order r of Grβ©L. Let L0=γaβ’bγ. Then |L0|=r, and L0β©N=1=L0β©Grβ©L. First assume that L0 is not normal in G. Then L0β€L0Gβ€Zπβ’(G). On the other hand, from the G-isomorphism Nβ’L/NβL, it follows that Lβ€Zπβ’(G) by Lemma 2.1 and Proposition 3.4. Moreover, Gr=L0β’(Grβ©L) since |N|=r and L0β©(Grβ©L)=1. Hence we have Nβ€Grβ€Zπβ’(G), and so Gβπ by Lemma 3.2β(1). This contradiction shows that Cβ©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, Nβ€Q and Nβ°Ξ¦β’(Q)β€Ξ¦β’(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 SchurβZassenhaus theorem. Moreover, p divides |M| by claim (3). Hence we have (4).

(5) M=γxγ is a cyclic π-abnormal Sylow p-subgroup of G, and every proper subgroup of M is π-normal in G:βBy claim (4), MβG/Dβπ, and p divides |M|. Therefore, every proper subgroup V of M is not π-abnormal in G, so V is π-normal in G by hypothesis. Let Mp be a Sylow p-subgroup of M. Assume Mpβ M. Then Mp is π-normal in G. Moreover, in this case, each Sylow subgroup P of M is π-normal in G, and P is a Sylow subgroup of G by claim (4). It follows that all Sylow subgroups of G are π-normal in G by claims (3) and (4), and therefore Gβπ by Proposition 3.4. This contradiction shows that M=Mp is π-abnormal in G and every proper subgroup of M is π-normal in G. Therefore, M is cyclic since the set of all π-normal subgroups of G forms a sublattice of the lattice of subgroups of G by Proposition 1.8. Hence we have (5).

(6) Zπβ’(G)=MG=γxpγ:βLet Z=Zπβ’(G), V=γxpγ. Assume Zβ©Dβ 1. Let U=(Zβ©D)β’M. Then Zβ©Dβ€Zβ©Uβ€Zπβ’(U) by Lemma 3.2β(2), so U/Zπβ’(U)βM/Mβ©Zπβ’(U)βπ, and hence Uβπ by Lemma 3.2β(1). But then M is not π-abnormal in G, contrary to claim (5). Therefore, Zβ©D=1, so Zβ€M, and hence, in fact, Zβ€VG. It is clear that Dβ€CGβ’(VG), so VGβ€Zβ’(G) by claim (5). Therefore, Vβ€VGβ€Zβ€V by Lemma 3.2β(1). Hence we have MG=VG=V=Zβ’(G)=Z.

(7) Myβ€NGβ’(H) for every proper subgroup H of D and every yβG:βLet V be a maximal subgroup of D such that Hβ€V. Then V is normal in G by claim (3). Let E=Vβ’My and D0=Eπ. It is clear that D0β€V. Moreover, D0β 1 since otherwise Eβπ, which implies that the subgroups My and M are not π-abnormal in G, contrary to claim (5). Furthermore, claim (1) implies E=D0βMy=VβMy, and every subgroup of D0=V is normalized by My. Hence Myβ€NGβ’(H).

(8) Every proper subgroup H of D is normal in G, and hence D is abelian:βClaim (7) implies MGβ€NGβ’(H), where M is π-abnormal in G by claim (5), and so G=MGβ€NGβ’(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.βIf Gβπ, then every subgroup of G is π-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 aβG, we have Maβ€A, so A is π-abnormal in G since the subgroups M and Ma are π-abnormal in G. Now assume that |M| does not divide |A|. Then, for a Sylow p-subgroup Ap of A and for some bβG, we have (Ap)b<M, so the subgroups (Ap)b and Ap are π-normal in G by hypothesis. Now note that A=(Aβ©D)β’Ap, where Aβ©D is normal in G, and so A is π-normal in G by Proposition 1.8. β

Communicated by Evgenii I. Khukhro

Award Identifier / Grant number: 11401264

Award Identifier / Grant number: PPZY 2015A013

Funding statement: Research is supported by an NNSF grant of China (Grant No. 11401264) and a TAPP of Jiangsu Higher Education Institutions (PPZY 2015A013).

## Acknowledgements

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

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