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Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) 117:105
https://doi.org/10.1007/s13398-023-01443-5
ORIGINAL PAPER
On two classes of generalised finite T-groups
Adolfo Ballester-Bolinches1,2 ·M. Carmen Pedraza-Aguilera3·
Vicent Pérez-Calabuig2
Received: 6 January 2023 / Accepted: 11 April 2023 / Published online: 21 April 2023
© The Author(s) 2023
Abstract
Let σ={σi:i∈I}be a partition of the set Pof all prime numbers. A subgroup Xof a
finite group Gis called σ-subnormal in Gif there is a chain of subgroups
X=X0≤X1≤ ··· ≤ Xn=G
where for every j=1,...,nthe subgroup Xj−1is normal in Xjor Xj/CoreXj(Xj−1)is
aσi-group for some i∈I. A group Gis said to be σ-soluble if every chief factor of Gis
aσi-group for some i∈I. The aim of this paper is to study two classes of finite groups
based on the transitivity of the σ-subnormality. Some classical and known results are direct
consequences of our study.
Keywords Finite group ·σ-soluble group ·σ-subnormal subgroup ·σ-nilpotency ·
Factorised group
Mathematics Subject Classification 20D10 ·20D20
1 Introduction and statements of results
All groups considered in this paper are finite.
BAdolfo Ballester-Bolinches
Adolfo.Ballester@uv.es
M. Carmen Pedraza-Aguilera
mpedraza@mat.upv.es
Vicent Pérez-Calabuig
Vicent.Perez-Calabuig@uv.es
1Department of Mathematics, Guangdong University of Education, Guangzhou 510310, Guangdong,
People’s Republic of China
2Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100 Burjassot, Valencia,
Spain
3Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Camí de Vera,
s/n, 46022 Valencia, Valencia, Spain
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105 Page 2 of 11 A. Ballester-Bolinches et al.
The springboards for the results of this paper are the classical characterisations of Gaschütz
and Robinson of the class of all T-groups, or groups in which every subnormal subgroup is
normal, and later developments of them.
Gaschütz proved the following characterisation of soluble T-groups in terms of their
normal structure.
Theorem 1 [1, Theorem 2.1.11], [2]Let G be a group and let L be the nilpotent residual of
G. Then, G is a soluble T -group if, and only if, the following conditions hold:
1. L is a normal abelian Hall subgroup of G with odd order;
2. G/L is a Dedekind group;
3. Every subgroup of L is normal in G.
Following Robinson (see [1, Definition 2.2.1]), we say that a group Gsatisfies the property
Cp(pa prime) if every subgroup of a Sylow p-subgroup Pof Gis normal in the normaliser
of NG(P). Robinson proved:
Theorem 2 [1, Theorem 2.2.2], [3]A group G is a soluble T -group if, and only if, G is a
Cp-group for all primes p.
Kegel (see [4, Definition 6.1.4]) introduced an extension of the subnormality to formation
theory that emerges naturally from the structural study of the groups.
Recall that if Fis a formation, a class of groups which is closed under taking epimorphic
images and subdirect products, then every group Ghas a smallest normal subgroup with
quotient in F. This subgroup is called the F-residual of Gand it is denoted by GF.Wesay
that the formation Fis subgroup-closed if HFis contained in GFfor all subgroups Hof a
group G.
Definition 1 Let Fbe a formation. A subgroup Uof a group Gis called K-F-subnormal in
Gif either U=Gor there is a chain of subgroups
U=U0<U1<··· <Un=G
such that Ui−1is either normal in Uior Ui−1is a maximal subgroup of Uiwith UF
i⊆Ui−1,
for i=1,2,...,n.
It is rather clear that the K-N-subnormal subgroups of a group G, for the formation Nof
all nilpotent groups, are exactly the subnormal subgroups of G. The reader is referred to [4,
Chapter 6] for a full discussion of the properties of these subgroups.
Bearing in mind Kegel’s extension of the subnormality within the framework of formation
theory and its strong impact on the subgroup structure, it seems interesting to study possible
extensions of Gaschütz and Robinson’s theorems to this context in order to help us to better
understand the role of the subnormality in the subgroup lattice.
Definition 2 [5, Definition 3] Let Fbe a subgroup-closed saturated formation. A group Gis
said to be a TF-group if every K-F-subnormal subgroup of Gis normal in G.
It is clear that TN=T. The following theorem confirms that a Gaschütz-type
characterization of soluble TF-groups holds.
Theorem 3 [5, Theorem 3] A group G is a soluble TF-group if, and only if, the following
conditions hold:
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1. GFis a normal abelian Hall subgroup of G with odd order;
2. X/XFis a Dedekind group for every X ≤G;
3. Every subgroup of GFis normal in G .
We would like to point up that every TF-group Gis in fact a T-group because every
subnormal subgroup of Gis K-F-subnormal in G.
On the other hand, Skiba [6] generalised the notions of solubility, nilpotency and subnor-
mality introducing the interesting concepts of σ-solubility, σ-nilpotency, and σ-subnormality,
in which σis a partition of the set P, the set of all primes. Thus, P=i∈Iσi, with σi∩σj=∅
for all i= j.
A group Gis said to be σ-primary if the prime factors, if any, of its order all belong to
thesamememberofσ.
Definition 3 A group Gis called σ-soluble if every chief factor of Gis σ-primary. Gis said
to be σ-nilpotent if it is a direct product of σ-primary groups.
If σis the minimal partition, that is, σ={{2},{3},{5},...}, then the class of σ-soluble
groups is just the class of soluble groups and the class of σ-nilpotent groups is just the class
of nilpotent groups. Furthermore, the class Nσof all σ-nilpotent groups is a subgroup-closed
saturated Fitting formation ([6, Corollary 2.4 and Lemma 2.5]).
Skiba(see[6, Theorem B] and [7]) proved that σ-soluble groups have a good arithmetic
behaviour with respect to the members of the partition σ.
Theorem 4 If G is a σ-soluble group, then G has a Hall σi-subgroup E and every σi-
subgroup of G is contained in a conjugate of E for all i ∈I . In particular, if i ∈I, theHall
σi-subgroups of G are conjugate. Furthermore, G has Hall σ
i-subgroups.
Definition 4 A subgroup Hof a group Gis called σ-subnormal in Gif there exists a chain
of subgroups
H=H0≤H1≤··· ≤ Hn=G,
where, for every i=1,...,n,Hi−1is normal in Hior Hi/CoreHi(Hi−1)is σ-primary.
The σ-subnormal subgroups play an important role in the study of soluble groups, and they
are exactly the subnormal subgroups for the minimal partition. Moreover, the σ-subnormal
subgroups of a σ-soluble group Gare exactly the K-Nσ-subnormal subgroups of G,and
so they are a sublattice of the subgroup lattice of G. Furthermore, by [4, Lemma 6.1.6], σ-
subnormality is a transitive subgroup embedding property, and epimorphic images preserve
σ-subnormal subgroups.
Zhang et al. [8] extended the class of all T-groups introducing the class of Tσ-groups.
Definition 5 A group Gis said to be a Tσ-group if every σ-subnormal subgroup of Gis
normal in G.
They showed σ-versions of Gaschütz and Robinson’s local characterisation of T-groups.
To this end, they considered the following extension of the class of all Cp-groups.
Definition 6 [8, Definition 1.8] Let i∈I. A group Gsatisfies condition Ciif every subgroup
Kof every Hall π-subgroup Hof G(π⊆σi) is normal in the normaliser NG(H).
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105 Page 4 of 11 A. Ballester-Bolinches et al.
Note that if Gsatisfies condition Ci,thenGsatisfies condition Cpfor all p∈σiand every
Hall σi-subgroup of Gis Dedekind.
The importance of the property Ciis underscored in the next result, which is a
characterisation of σ-soluble Tσ-groups.
Theorem 5 [8, Theorem 1.10] Let G be a σ-soluble group. Let D =GNσ. Then, the following
statements are equivalent.
1. GisaT
σ-group;
2. G satisfies condition Cifor all i .
3. G satisfies the following conditions:
(a) G=DM, where D is an abelian Hall subgroup of G of odd order and M is a
Dedekind group;
(b) every element of G induces a power automorphism on D; and
(c) Oσi(D)has a normal complement in a Hall σi-subgroup of G for all i .
It is important to stress that the σ-soluble Tσ-groups are just the TF-groups for the
subgroup-closed formation F=Nσof all σ-nilpotent groups. Moreover, every Tσ-group is
aT-group.
The key to help us to understand the difference between σ-soluble Tσ-groups and T-groups
is the following theorem.
Theorem A Aσ-soluble group G is a Tσ-group if, and only if, G is a soluble T -group and
the Hall σi-subgroups of G are Dedekind for all i ∈I.
We would like to point out that Theorem 5is a direct consequence of Theorem A.
An interesting and smart extension of normality to saturated formations in the soluble
universe was given by Doerk and was introduced and studied for the first time in [9]. The
definition depends on the minimal local definition of the saturated formation, and can be also
considered as a subgroup embedding property in the general finite universe.
Definition 7 [9, Definition 3.1] Let Fbe a non-empty saturated formation and let Gbe a group.
A subgroup Aof Gis called F-Dnormal in Gif Asatisfies the following two conditions:
1. σ(|G:A|)⊆Char(F)={p:pprime and Cp∈F}.
2. If p∈Char(F),then[Ap
G,Af(p)]≤A,where fdenotes the smallest local definition of
Fas local formation, and Ap
G=Gp∈Sylp(G):Gp∩A∈Sylp(A).
The F-Dnormal subgroups turn out to be important in the study of the structure of the
groups (see [9,10]).
We are interested here in the Nσ-Dnormality. In order to keep our notation homogeneous,
we shall say that a subgroup Aof a group Gis σ-normal in Gif Ais a Nσ-Dnormal subgroup
of G.
Note that the smallest local definition of Nσis given by
f(p)=(1), if σi={p},
Eσi,if p∈σi,|σi|≥2,(1)
where Eσidenotes the class of all σi-groups.
As it was noted in [10, Remark 3.6], if Xand Hare subgroups of a group Gand πis a
set of primes, then [X,Oπ(H)]≤Hif, and only if, X≤NG(Oπ(H)). Therefore, the next
useful characterisation of σ-normal subgroups holds.
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Proposition 6 Let A be a subgroup of G and p a prime number. Then, A is a σ-normal
subgroup of G if, and only if, the following statements hold:
1. If p ∈σiand σi={p},thenG
p≤NG(A)for every G p∈Sylp(G)such that
Gp∩A∈Sylp(A).
2. If p ∈σiwith |σi|≥2and G p∈Sylp(G)such that G p∩A∈Sylp(A),then
Gp≤NG(Oσi(A)).
It is clear that every normal subgroup is σ-normal, and if σis the minimal partition, then
every σ-normal subgroup is normal.
Furthermore, arguing as in [11, Lemma 1.4], it follows that σ-normality is inherited in
intermediate subgroups, and if Ais a σ-normal subgroup of Gand Nis normal in G,then
AN/Nis σ-normal in G/N.Conversely,ifN≤Aand A/Nis σ-normal in G/N,thenA
is σ-normal in G. These facts will be used in the sequel without further reference.
The above basic properties of σ-normal subgroups yield the following.
Proposition 7 Every σ-normal subgroup of a σ-soluble group is σ-subnormal.
Proof Assume that Ais a σ-normal subgroup of a σ-soluble group G.Weshowthat Ais
σ-subnormal in Gby induction on |G|.Let Nbe a minimal normal subgroup of G. Then,
AN/Nis σ-normal in G/Nand so AN/Nis σ-subnormal in G/N. In particular, AN is
σ-subnormal in G. Assume that AN is a proper subgroup of G. Then, Ais σ-subnormal in
AN by induction. By transitivity, Ais σ-subnormal in G. Consequently, we may assume that
G=AN and CoreG(A)=1. Since Gis σ-soluble, it follows that Nis a σi-group for some
i∈I.Ifσi={p},thenNis a p-group and so Nnormalises A. In this case, Ais normal in G
and so Ais σ-subnormal in G. Assume that |σi|≥2, and let p∈σi.IfGp∈Sylp(G)such
that Gp∩A∈Sylp(A),thenGp∩N∈Sylp(N)and Gp≤NG(Oσi(A)). Therefore, for
every p∈π(N), there exists a Sylow p-subgroup of Nnormalising Oσi(A). In particular,
Nnormalises Oσi(A)andthenO
σi(A)is normal in G.Since Ais core-free in G, it follows
that Oσi(A)=1andsoGis a σi-group. In this case, Ais obviously σ-subnormal in G.
The following class of groups naturally emerges from the above proposition.
Definition 8 Aσ-soluble group Gis said to be a σT-group,ifeveryσ-subnormal subgroup
of Gis σ-normal in G.
Note that every Tσ-group is in fact a σT-group. However, these two classes of groups are
different because there are σT-groups which are not T-groups. For instance, if Gdenotes the
alternating group of degree 4, and σ={{2,3},{2,3}},thenGis an example of a σT-group
which is not a Tσ-group since it is not a T-group. Furthermore, the good behaviour of the
σ-normal and σ-subnormal subgroups with respect to epimorphic images implies that the
class of all σT-groups is closed by taking quotients.
We point up that the class of all σT-groups is just the class of all T-groups when σis the
minimal partition.
We say that a group Gis a σ-Dedekind group if every subgroup of Gis σ-normal in G.
It is clear that the class of all σ-Dedekind groups is a subclass of the class of all σT-groups.
The second main result of the paper is a σ-version of Theorem 1characterising the
σ-soluble σT-groups.
Theorem B Let G be a σ-soluble group and D =GNσ. Then, G is a σT-group if, and only
if, G satisfies the following statements:
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105 Page 6 of 11 A. Ballester-Bolinches et al.
1. G=DM, where D is an abelian Hall subgroup of odd order;
2. if p ∈π(D)∩σi,forsomei ∈I , then D contains a Hall σi-subgroup of G;
3. Misaσ-Dedekind group;
4. every element of G induces a power automorphism of D.
The following generalisation of supersolubility is due to Guo, Chi and Skiba ([12])
Definition 9 A group Gis called σ-supersoluble if every chief factor of Gbelow the σ-
nilpotent residual of Gis cyclic.
Of course σ-supersolubility coincides with supersolubility for the minimal partition σ.
An important consequence of Theorem 1is that the class of all soluble T-groups is a
subgroup-closed class of supersoluble groups. For σT-groups, the following extension holds.
Corollary 8 The class of all σ-soluble σT-groups is a subgroup-closed subclass of the class
of all σ-supersoluble groups.
2 Proofs of Theorem Aand Theorem 5revisited
The proof of Theorem Adepends on the following.
Lemma 9 Let G be a σ-soluble Tσ-group. Then, G is soluble.
Proof We argue by induction on |G|. Since the class of all σ-soluble Tσ-groups is closed
under taking homomorphic images by [4, Lemma 6.1.6] and soluble groups are a saturated
formation, we may assume that Gis a primitive group with a unique minimal normal sub-
group, Nsay, and G/Nsoluble. Since Gis σ-soluble, it follows that Nis σ-primary. Let A
be a subgroup of prime order of N. Then, Ais σ-subnormal in N.By[4, Lemma 6.1.6], A
is σ-subnormal in G.SinceGis a Tσ-group, we have that Ais normal in G. The minimality
of Nyields N=Aand Nis abelian. Consequently, Gis soluble.
Proof of Theorem AAssume that Gis a σ-soluble Tσ-group. Then, by Lemma 9,Gis a
soluble T-group. By Theorem 3,everyσ-nilpotent subgroup of Gis a Dedekind group.
Consequently, every Hall σi-subgroup of Gis Dedekind for all i∈I.
Conversely, assume that Gis a soluble T-group and the Hall σi-subgroups of Gare
Dedekind for all i∈I.LetZbe the σ-nilpotent residual of G. Then, G/Zis a direct product
of Hall σi-subgroups of G/Z,i∈I, which are Dedekind. In particular, G/Zis a direct
product of nilpotent subgroups and then G/Zis nilpotent. Consequently, Zis actually the
nilpotent residual of G. By Theorems 1and 3,Gis a Tσ-group.
We now apply Theorem Ato give an alternative shorter proof of Theorem 5.
Proof of Theorem 5If Gisaσ-soluble Tσ-group and i∈I, then the Hall σi-subgroups of
Gare Dedekind by Theorem A. Assume that π⊆σiand let HbeaHallπ-subgroup of G.
Then, His also a Dedekind group and every subgroup Kof His subnormal in NG(H).Since
NG(H)is a T-group, it follows that Kis normal in NG(H). Hence, Gsatisfies condition Ci
and Statement (1) implies Statement (2).
Assume now that Statement (2) holds. Then, Gsatisfies Cpfor all p. Then, Gis a soluble
T-group by Theorem 2, and every Hall σi-subgroup of Gis Dedekind. By Theorem A,Gis
aσ-soluble Tσ-group and Statement (2) implies Statement (1).
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Clearly, Statement (1) implies Statement (3) because every soluble T-group with Dedekind
Hall σi-subgroups for all i∈Isatisfies Statement (3) (note that in this case the nilpotent
and σ-nilpotent residuals of Gcoincide, and we can apply Theorem 1). Assume now that
Statement (3) holds. Then, Dis just the nilpotent residual of Gbecause Mis a Dedekind
group. Therefore, by Theorem 1,Gis a soluble T-group. Note that if His a Hall σi-subgroup
of G,thenH=Oσi(D)×Xfor a subgroup Xof H.Since His a T-group and Xis nilpotent,
it follows that Xis a Dedekind group. Note that Oσi(D)is also Dedekind. Since Oσi(D)is
of odd order, it follows that His a Dedekind group. By Theorem A,Gis a Tσ-group and
Statement (1) holds.
3 The proof of Theorem B
We start by noting that if Ais a σ-normal subgroup of a group Gand Ais a σi-group for some
i∈I,theneverySylow p-subgroup of G, for every prime p/∈σi, normalises A. Therefore,
Oσi(G)≤NG(A). This fact will be also used in the sequel without further reference.
Proof of Theorem BAssume that Gisaσ-soluble σT-group and let Dbe the σ-nilpotent
residual of G.WeprovethatGsatisfies Statements (1), (2), (3) and (4) by induction on |G|.
We split the proof into the following steps.
Step 1. D is nilpotent.
Assume that Dis not nilpotent. If Lis a minimal normal subgroup of G,wehavethat
G/Lis a σT-group and DL/Lis the σ-nilpotent residual of G/L. The minimal choice of G
implies that DL/Lis nilpotent. Since the class of all nilpotent groups is a saturated formation,
it follows that Ghas a unique minimal normal subgroup, Asay. Furthermore, Ais contained
in D,D/Ais nilpotent and A∩(G)=1. Since Gis σ-soluble, it follows that Ais a
σi-group for some i∈I.
Let Bbe a subgroup of A. Then, Bis σ-subnormal in A. Consequently, Bis σ-subnormal
in Gand so Bis σ-normal in Gbecause GisaσT-group. Now, D≤Oσi(G)≤NG(B).This
means that Ais acted on by conjugation by Das a power automorphisms. In particular, A
is abelian, CG(A)=A=F(G),andDis supersoluble. Let pbe the largest prime dividing
|D|. Then, Op(D)is a Sylow p-subgroup of Dand it is a normal subgroup of G. Therefore,
A=Op(D).
Let Pbe a Sylow p-subgroup of G. Then, X=DP is σ-subnormal in Gand so Xis a
σT-group. If G=DP, then every subgroup of A∩Z(P)= 1 would be normal in Gand
so Awould be cyclic of prime order. In this case Awould be central in Dand Dwould be
nilpotent, contrary to supposition. Hence, Xis a proper subgroup of G. Then, by induction,
the σ-nilpotent residual X∗of Xis an abelian normal Hall subgroup of Xcontained in D.
Since Ais self-centralising in X, it follows that X∗is a p-group and X∗is a Sylow p-subgroup
of G. Therefore, DP =D. In particular, A=Pis the unique Sylow p-subgroup of G.
Note that Z=Oσi(G)normalises every subgroup of A. Therefore, Zis a proper subgroup
of G.Since ZisaσT-group and contains D, we have that the σ-residual Z∗of Zis normal in
Gand contained in A. Therefore, Z∗=A. Assume that Y=Oσj(G)is a proper subgroup of
G. Arguing as above, we have that Y∗=A.SinceG=YZ,andY/Aand Z/Aare normal
σ-nilpotent subgroups of G/A, it follows that G/Ais σ-nilpotent and D=A, contrary to
supposition. Consequently, G=Oσj(G)for every j= iand so D=Oσi(G).
Let Mbe a core-free maximal subgroup of Gsuch that G=AM and A∩M=1. Then,
Mis a σT-group. By induction, the σ-nilpotent residual M∗of Mis an abelian Hall subgroup
of Mand there exists a σ-Dedekind subgroup Tof Msuch that M=M∗Tand M∗∩T=1.
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Clearly, D=AM∗. Assume that q∈σi∩π(D)and let Qbe a Sylow q-subgroup of D.
Then, AQ is a subnormal σi-subgroup of G. Hence, Qis σ-subnormal in G.SinceGis a
σT-group, it follows that Qis σ-normal in G. Thus, Dnormalises Qand so Q≤F(D)=A.
Hence, q=pand σi∩π(D)={p}. This implies that AT is a Hall σi-subgroup of Gand
M∗is a Hall σ
i-subgroup of G.
Observe that Dis a proper subgroup of Gbecause Gis σ-soluble. Let Xbe a proper
subgroup of Gcontaining D.ThenXis σ-subnormal in Gand so Xis a σT-group. The
induction hypotheses guarantee that X∗is an abelian Hall subgroup of Xand every subgroup
of X∗is normal in X.If X∗=1, then Xis σ-nilpotent and so is D.Then Dis a σi-group
because CG(A)=Aand so D=A. Hence Dis nilpotent. We may assume that X∗= 1.
Then X∗∩A= 1andso A≤X∗because Ais a Sylow p-subgroup of G. Consequently
D∗=X∗=A. Applying Maschke’s Theorem [13,A; Theorem 11.5], and [13, B; Proposition
9.3], Ais a direct product of G-isomorphic cyclic minimal normal subgroups of X.LetA0
one of them. Then CX(A0)=CX(A)=Aand X/Ais cyclic of order dividing p−1. Hence
every Hall p-subgroup of Xis cyclic.
Let Sbe a subgroup of Tof prime order. Assume E=DS is a proper subgroup of G.
Then E∗=A,andthen AS is a σ-subnormal subgroup of Ewhich is σ-subnormal in G.
Since AS is a σi-group, we have that Sis σ-subnormal in G.SinceGis a σT-group, it follows
that Sis σ-normal in Gand so Anormalises S. This contradiction yields T=1andG=D,
and this is a contradiction.
Assume G=DS.IfCD(S)= 1, then there exists a prime rand a Sylow r-subgroup Rof
Dsuch that CR(S)= 1. Assume that r=p.ThenCA(S)is a normal subgroup of G.Since
Ais a minimal normal subgroup of G, it follows that CA(S)=Aand so S≤CG(A)=A,a
contradiction. Hence r= p.Let ZbeaHallr-subgroup of D.Then A≤Zand Zis normal
in G.ThenRZ/Zis cyclic and CRZ/Z(SZ/Z)= 1. Hence CRZ/Z(SZ/Z)=RZ/Zby [13,
A; 12.5]. Thus G/Z=RZ/Z×SZ/Zand G/Zis nilpotent. Hence D≤Z, a contradiction.
Hence CD(S)=1. In particular, Gis a Frobenius group with kernel Dand complement S.
Applying [14, 10.5.6], Dis nilpotent, the final contradiction.
Step 2. D is a Hall subgroup of G.
We may assume that D= 1. Let Nbe a minimal normal subgroup of Gcontained in D.
Since G/Nis a σT-group, we can apply induction to conclude that D/Nis a Hall subgroup
of G. Hence (|D/N|,|G/N:D/N|)=1. Now, Nis a p-group for some prime pbecause D
is nilpotent. Assume that p∈π(D/N). Then, (|D|,|G:D|)=1and Dis a Hall subgroup
of G. Consequently, we may assume that Nis a Sylow p-subgroup of D.
Let Fbe the normal Hall p-subgroup of Dand assume that F= 1. Let Abe a
minimal normal subgroup of Gcontained in F. Then, π(A)⊆π(D/N). By induction,
(|D/A|,|G/A:D/A|)=1=(|D|,|G:D|)and Dis a Hall subgroup of G. Conse-
quently, we may assume that D=Nis a minimal normal subgroup of G. According to [4,
Theorem 4.2.17], G=DM and D∩M=1.
Assume that p∈σifor some i∈I.LetGibeaHallσi-subgroup of G. Then, Gi=
D(Gi∩M). Note that Gi∩Mis a σ-subnormal subgroup of Gand so Gi∩Mis σ-normal
in G. Hence, Dnormalises Gi∩Mand so Gi∩Mis a normal subgroup of Gi. In particular,
D≤Z(Gi). Then, every subgroup Zof Dis a σ-normal subgroup of Gnormalised by Gi.
Therefore, G=Oσi(G)Gi≤NG(Z)and Zis normal in G. This implies that |D|= p.
Clearly, we may assume that Gi∩M= 1. Let Lbe a minimal normal subgroup of G
contained in Gi∩M.LetPbeaSylow p-subgroup of G. Then, by induction, DL/Lis a
Sylow p-subgroup of G/Land so PL =DL and P=D×(L∩P).NotethatPis σ-normal
in G.
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Let Ebe a Hall σi-subgroup of G.Since Pis σ-normal in G, it follows that Enormalises
P.LetV=PE. Note that Vis a σT-group. The σ-nilpotent residual V∗of Vis contained
in DandsoeitherVis σ-nilpotent or D=V∗.
Assume that Vis σ-nilpotent. Then, Eand Gicentralise D. Hence, D≤Z(G)and so G
is σ-nilpotent. Consequently, we may assume that D=V∗.IfV<G,thenDis a Sylow
p-subgroup of Vby induction. Hence, D=Pis a Sylow p-subgroup of G. Consequently,
we may assume that V=Gand then P=Gi=D×L. In this case, every subgroup
of Pis normal in G.Accordingto[1, Lemma 2.1.3], Dand Lare G-isomorphic. Since
Op(G)≤CG(LD/D)=CG(D), it follows that D≤Z(G). Therefore, Gis σ-nilpotent
and D=1 is a Hall subgroup of G.
The arguments used in the proof of Step 2 show
Step 3. If p ∈π(D)∩σi,then D contains a Hall σi-subgroup of G .
Step 4. Let H be a σi-subgroup of D.Then, H is normal in G.
We may assume that H= 1. Let Abe a minimal normal subgroup of Gcontained in D.
Since every subgroup of the σ-residual D/Aof G/Ais normal in G/Aby induction, it follows
that HAis normal in G.IfAis a σi-group, then His normal in HAbecause Dis nilpotent.
This implies that His characteristic in HAand so His normal in G. Consequently, we may
assume that Dis a σi-group. In this case, His σ-normal in Gbecause it is σ-subnormal in
Gand Gis a σT-group. Then, D≤Oσi(G)normalises H.LetGibeaHallσi-subgroup of
G.Since Dis a Hall subgroup of Gi,thenDis complemented in Giby a subgroup Fof Gi.
In addition, Fis σ-normal in Gbecause Giis σ-normal in G. Consequently, Dnormalises
Fand so F≤CG(D). Thus, Ginormalises Hand so G=GiOσi(G)≤NG(H)and His
normal in G.
Since every subgroup of Dis a direct product of σi-subgroups of Gfor all i∈I,wehave
Step 5. Every subgroup of D is normal in G.
Step 6. D is abelian.
By Step 4, Dis a Dedekind group. Let Mbeaσ-nilpotent normaliser of G. Then,
G=DM.LetD2be the Sylow 2-subgroup of D. Then, D2is normal in Gand every chief
factor of Gbelow D2is central in G. In particular, D2is contained in Z∞(G), the hypercentre
of G.By[4, Proposition 4.2.6], D2is contained in Mand so G=D2M.SinceD2is normal
in Gand G/D2is σ-nilpotent, it follows that D≤D2. Consequently, Dis of odd order
and Dis abelian by [14, 5.3.7].
Conversely, assume that Gsatisfies Statements (1), (2), (3) and (4). We prove that Gis
aσT-group by induction on |G|. We may assume that D= 1. It is clear that the conditions
on Ghold in every epimorphic image of G. Consequently, G/Nis a σT-group for every
non-trivial normal subgroup Nof G.
Let Hbeaσ-subnormal subgroup of G.Weshowthat His σ-normal in G.Wemayassume
that H<G. Assume that CoreG(H)= 1. Then, H/CoreG(H)is a σ-subnormal subgroup
of the σT-group G/CoreG(H). Then, H/CoreG(H)is σ-normal in G/CoreG(H)and then
His σ-normal in G. Hence, we may assume that CoreG(H)=1. Since every subgroup of
Dis normal in G,wehavethatD∩H=1. Then, His contained in a conjugate of Min
G. Without loss of generality, we may assume that H≤M. Furthermore, since every Hall
σt-subgroup of His normal in Hfor all t∈I, we may assume that His a σi-subgroup of
Gfor some i∈I. By Statement (2), Dis a σi-group.
Note that Mcannot be σ-subnormal in Gsince Mis a proper subgroup of Gwhich cannot
be contained in a σ-subnormal maximal subgroup of G. Consequently, DH <Gand then
DH satisfies the hypotheses of the theorem. By induction, DH is a σT-group and so His
σ-normal in DH. In this case, D=Oσi(G)≤NG(H).
123
105 Page 10 of 11 A. Ballester-Bolinches et al.
Let pbe a prime and let GpbeaSylow p-subgroup of Gsuch that Gp∩His a Sylow
p-subgroup of H.If p∈π(D),thenGpnormalises H. Suppose that p∈π(M),then
there exists d∈Dsuch that Gd
p≤M, then either Gd
pnormalises Hor Gd
pnormalises
Oσj(H). Hence, either Gpnormalises Hd=Hor Gpnormalises (Oσj(H))d=Oσj(H).
Consequently, His a σ-normal subgroup of G.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
These results are a part of the R+D+i project supported by the Grant PGC2018-095140-B-I00, funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. The research of the first
author is also supported by the Grant 12071092, funded by the National Science Foundation of China. The
authors thank the referees for their careful reading of the paper.
Data availability Data sharing not applicable to this article as no datasets were generated or analysed during
the current study.
Declarations
Conflict of interest The authors have no competing interests to declare that are relevant to the contents of this
article.
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