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Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2022) 116:74
https://doi.org/10.1007/s13398-022-01215-7
ORIGINAL PAPER
Carter and Gaschütz theories beyond soluble groups
Milagros Arroyo-Jordá1·Paz Arroyo-Jordá1·Rex Dark2·Arnold D. Feldman3·
María Dolores Pérez-Ramos4
Received: 19 May 2021 / Accepted: 19 January 2022 / Published online: 11 February 2022
© The Author(s) 2022
Abstract
Classical results from the theory of finite soluble groups state that Carter subgroups, i.e.
self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent
covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble
groups. This paper presents an extension of these facts to π-separable groups, for sets of
primes π, by proving the existence of a conjugacy class of subgroups in π-separable groups,
which specialize to Carter subgroups within the universe of soluble groups. The approach
runs parallel to the extension of Hall theory from soluble to π-separable groups by ˇ
Cunihin,
regarding existence and properties of Hall subgroups.
Keywords Finite soluble groups ·π-separable groups ·Carter subgroups ·Hall systems
Mathematics Subject Classification 20D10 ·20D20
BMaría Dolores Pérez-Ramos
Dolores.Perez@uv.es
Milagros Arroyo-Jordá
marroyo@mat.upv.es
Paz Arroyo-Jordá
parroyo@mat.upv.es
Rex Dark
rex.dark@nuigalway.ie
Arnold D. Feldman
afeldman@fandm.edu
1Escuela Técnica Superior de Ingeniería Industrial, Instituto Universitario de Matemática Pura y
Aplicada IUMPA, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
2School of Mathematics, Statistics and Applied Mathematics, National University of Ireland,
University Road, Galway, Ireland
3Franklin and Marshall College, Lancaster, PA 17604-3003, USA
4Departament de Matemàtiques, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot,
València, Spain
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74 Page 2 of 13 M. Arroyo-Jordá et al.
1 Introduction
All groups considered are finite.
The well-known result of Carter [7] states that each soluble group possesses exactly one
conjugacy class of self-normalizing nilpotent subgroups (the so-called Carter subgroups).
His discovery was of interest at the time not only by analogy with Cartan subalgebras,
but also by sharing the flavour of Hall’s result on the existence and conjugacy of Hall ρ-
subgroups, for all sets of primes ρ, in every finite soluble group. Indeed, classical results of
Hall theory state that soluble groups are characterized by the existence of Hall ρ-subgroups
for all sets of primes ρ(Hall [15,16]). Cleverly and beautifully, Gaschütz [12] unifies both
families of subgroups under the concept of F-covering subgroups associated to saturated
formations F, by considering the class Sρof soluble ρ-groups, and the class Nof nilpotent
groups, respectively, as very particular cases of saturated formations. This is the origin of
the theory of distinguished conjugacy classes of subgroups in finite soluble groups related to
certain classes of groups, which quickly splits into the theories of covering subgroups and
projectors related to Schunck classes and formations, and the dual theory of injectors and
Fitting classes. (See Definitions 3.1,3.2.) We refer to the excellent monographs [5,9]for
an account of developments on the topic in the universes of soluble and finite groups. On
the other hand, if πis a set of primes, π-separable groups have Hall π-subgroups, and also
every π-subgroup is contained in a conjugate of any Hall π-subgroup, by a well-known result
of ˇ
Cunihin [6]. However, there has been no successful extension of Carter and Gaschütz’
results to π-separable groups corresponding to the extension of Hall’s results to π-separable
groups by ˇ
Cunihin. The aim of this paper is to provide such an extension. Note that for
non-soluble groups, the known equivalence of Carter subgroups, N-covering subgroups and
N-projectors no longer holds. In arbitrary finite groups N-projectors do always exist (cf. [9,
III. Theorem (3.10)]), though they do not form a conjugacy class of subgroups in general, and
the existence of N-covering subgroups is not guaranteed. For Carter subgroups no counterpart
has been found for a general non-soluble group, though they are conjugate when existing,
as finally settled by Vdovin in [19]. Within the theory of F-normalizers, initiated by Carter
and Hawkes, there appear extensions of classical embedding properties of subgroups, such
as F-subnormal subgroups, associated to saturated formations F(cf. [8,17], [9, Chapter V],
[5, Chapters 4, 6]). In [2] a definition of F-normality in soluble groups consistent with the
lattice properties of F-subnormal subgroups is achieved. This concept is applied in this paper
to extend Carter subgroups taking heed of its very definition, as nilpotent self-normalizing
subgroups, and enables us to address the lack of a Carter and Gaschütz counterpart in the
extension of Hall theory from soluble groups to π-separable groups. The deep knowledge
of techniques and progress now achieved in this area allow us to reach the results of this
paper. We notice that a group Gis soluble if and only if Gis ρ-separable for all sets of
primes ρ. It is an easy but motivating observation that a group is π-separable if and only
if it is ρ-separable for suitable sets of primes related to the set of primes π, as stated in
Proposition 2.1. It is then straightforward to provide an extension of the existence of Hall
systems of soluble groups to π-separable groups, as shown in Sect. 2. At the end of that
section we discuss to what extent the existence of these extended Hall systems characterizes
π-separability. This setting suggests the consideration of classes Nπof groups which are a
direct product of a π-group by a nilpotent π-group, where πstands for the complement
of πin the set Pof all prime numbers, to play the role of the class of nilpotent groups.
In Sect. 3, we present briefly the general framework from the theory of soluble groups and
classes of groups which relates to our purposes, focusing on some concepts and preliminary
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Carter and Gaschütz theories beyond... Page 3 of 13 74
results which are crucial to establish our main results; particularly, we introduce and adjust
the above-mentioned extension of normality for saturated formations according to our needs.
The main results of the paper are then carried out in Sect. 4. For a set of primes π,we
prove that if Gis a π-soluble group, then the Nπ-projectors coincide with the Nπ-covering
subgroups and they form a non-empty conjugacy class of subgroups (Theorem 4.5). (Notice
that, by the Feit-Thompson theorem, for any set of primes ρ,aρ-separable group is either a
ρ-soluble group or a ρ-soluble group, so that the hypothesis of π-solubility means no real
restriction but more a question of adjusting the sets of primes; see Remark 4.1). Besides,
Theorem 4.14 proves that in a π-soluble group, Nπ-projectors, and so also Nπ-covering
subgroups, can be described and characterized as a family of subgroups which specialize to
Carter subgroups within the universe of soluble groups. Finally, it is remarkable that Carter
subgroups are the cornerstone for the existence and conjugacy of injectors associated to
Fitting classes in soluble groups. In a forthcoming paper [4], our Carter-like subgroups are
used to generalize these results to π-separable groups.
2 From soluble to -separable groups
As mentioned in the Introduction, we pursue an extension of the theory of finite soluble
groups to the universe of π-separable groups, πa set of primes. With this aim we analyze
first the reach of π-separability further from the universe of soluble groups. We refer to [14]
for basic results on π-separable groups, and to [9] for background on classes of groups;
we shall adhere to their notations. If πis a set of primes, let us recall that a group Gis
π-separable if every composition factor of Gis either a π-group or a π-group. We start by
noticing that a group Gis soluble if and only if it is ρ-separable for all sets of primes ρ.
Regarding π-separability, πa set of primes, it is clearly equivalent to π-separability, so that
there is no loss of generality to assume that 2 ∈π. Then, by the Feit-Thompson theorem,
aπ-separable group is π-soluble, i.e. the group is π-separable with every π-composition
factor a p-group for some prime p∈π. The following extension for π-separable groups is
easily proved:
Proposition 2.1 For a group G, if 2∈π⊆Pthe following statements are pairwise equiva-
lent:
1. Gisπ-separable;
2. Gisρ-separable for every set of primes ρsuch that either π⊆ρor π∩ρ=∅;
3. Gisπ-separable (π-soluble).
Remark 2.2 The need of the hypothesis 2 ∈πfor the validity of the equivalences in Propo-
sition 2.1 is clear. Certainly, for a group Gand any set of primes π, it holds that 2 →1↔3.
But statement 2 implies that the π-compositions factors of the group Gare soluble. Then,
by the Feit-Thompson theorem, if 2 /∈π, statement 2 is equivalent to the solubility of the
group Gand 1 2 in general.
Consequently, by Proposition 2.1,if2∈π⊆P,everyπ-separable group possesses a
Sylow p-complement of G, i.e. a Hall p-subgroup of G, for each p∈π,aswellasa
Hall π-subgroup, and these are pairwise permutable subgroups with coprime indices in the
group. In analogy with [9, I. Definitions (4.1), (4.5), (4.7)] it appears to be natural now to
introduce the following concepts, which are proven to hold in π-separable groups if 2 ∈π,
by the previous comment and as explained below:
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Definition 2.3 Let Gbe a group and πbe a set of primes.
Kπ:Acomplement π-basis of Gis a set Kπcontaining exactly one Sylow p-complement
of G, i.e. a Hall p-subgroup of G, for each p∈π, and exactly one Hall π-subgroup.
π:AHall π-system of Gis a set πof Hall subgroups of Gsatisfying the following two
properties:
1. For each set of primes ρsuch that either π⊆ρor π∩ρ=∅,πcontains exactly
one Hall ρ-subgroup.
2. If H,K∈π,thenHK =KH.
Bπ:ASylow π-basis of Gis a set Bπof subgroups of Gsatisfying the following two
properties:
1. Bπcontains exactly one Hall π-subgroup, and exactly one Sylow p-subgroup for
each p∈π.
2. If H,K∈Bπ,thenHK =KH.
Obviously these systems may not exist in arbitrary groups. By the previous comments, if
the group is π-separable and 2 ∈π, then complement π-bases do exist.
Note also that Sylow π-systems and complement and Sylow π-bases are hereditary with
respect to normal subgroups and factor groups.
For any set ρof primes and a group G,wedenotebyHall
ρ(G)the set of all Hall ρ-
subgroups of G.Ifpis a prime, then Syl p(G)will denote the set of all Sylow p-subgroups
of G. We keep mimicking the exposition in [9, I. Sect. 4]. The arguments there are easily
adapted to prove the following corresponding results.
Proposition 2.4 [9, I. Proposition (4.4)] Assume that the group G has a complement π-basis,
say Kπ(particularly, if the group G is π-separable and 2∈π⊆P). If ρis a set of primes
such that π⊆ρ,letG
ρ={X|X∈Hallp(G)∩Kπ,p∈ρ⊆π}. On the other hand,
if ρis a set of primes such that π∩ρ=∅,letG
ρ={X|(X∈Hallp(G)∩Kπ,p∈
ρ∩π)∨(X∈Hallπ(G)∩Kπ)}.Then
1. π:= {Gρ|(π ⊆ρ⊆P)∨(ρ ⊆P,ρ∩π=∅)}is a Hall π-system of G, and
2. πis the unique Hall π-system of G containing Kπ.
We shall say that πis the Hall π-system generated by the complement π-basis Kπ.
Corollary 2.5 [9, I. Corollary (4.6)] Let G be a π-separable group, 2∈π⊆P. Then there is
a bijective map between the set of all complement π-bases and the set of all Hall π-systems
of G, such that to each complement π-basis corresponds the Hall π-system generated by it,
and conversely, to each Hall π-system corresponds the complement π-basis contained in it.
On the other hand, it is clear that every Hall π-system contains a unique Sylow π-basis.
Also, each Sylow π-basis generates a unique Hall π-system, by taking the product of the
suitable elements in the basis to construct each element in the Hall π-system. We can easily
state also the following result:
Corollary 2.6 [9, I. Lemma (4.8)] Let G be a π-separable group, 2∈π⊆P. Then there is a
bijective map between the set of all Hall π-systems and the set of all Sylow π-bases of G, such
that to each Hall π-system corresponds the Sylow π-basis contained in it, and conversely,
to each Sylow π-basis corresponds the Hall π-system generated by it as described above.
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Carter and Gaschütz theories beyond... Page 5 of 13 74
If Kπis a complement π-basis of a group Gand g∈G, it is clear that Kg
π:= {Xg|X∈
Kπ}is again a complement π-basis of G, and this defines an action by conjugation of Gon
the set of all complement π-bases of G. Analogously, any group Gacts by conjugation on
the set of all its Hall π-systems, and also on the set of all its Sylow π-bases.
By Proposition 2.1 again we have that if Gis a π-separable group, 2 ∈π⊆P,thenG
acts transitively on the set of all Hall p-subgroups for every p∈π,aswellasonthesetof
all Hall π-subgroups, and the following result also holds.
Theorem 2.7 [9, I. Theorems (4.9), (4.10), (4.11), Corollary (4.12)] Let G be a π-separable
group, 2∈π. Then:
1. The number of Hall π-systems of G is S∈Kπ|G:NG(S)|,whereKπis a complement
π-basis of G.
2. The group G acts transitively by conjugation on the set of all complement π-bases, on
the set of all Hall π-systems, as well as on the set of all Sylow π-bases.
One might wish that the existence of Hall π-systems in finite groups would characterize
π-separability, but this is not the case, even assuming the transitive action of the group by
conjugation on the set of all Hall π-systems. The alternating group of degree 5 together
with the set π={2,3}is a counterexample. A characterization of π-separability by the
existence of Hall subgroups had been in fact given by Du [10], as shown in the next result.
For notation, for a group Gand any set of primes ρ, the group Gis said to satisfy Eρif G
has a Hall ρ-subgroup; if in addition each ρ-subgroup is contained in the conjugate of a Hall
ρ-subgroup, it is said that Gsatisfies Dρ.
Theorem 2.8 [10, Theorems 1, 3] For a group G and a set of primes π, the following
statements are pairwise equivalent:
(i) Gisπ-separable.
(ii) G satisfies:
1. Eπand Eπ;
2. Eπ∪{q}and Eπ∪{ p}, for all p ∈π,q ∈π.
(iii) G satisfies:
1. Eπand Eπ;
2. E{p,q}, for all p ∈π,q ∈π.
We point out finally that the existence of Hall π-systems together with π-dominance do
characterize π-separability, as we prove next.
Remark 2.9 We notice that for any group Gand any π⊆P, the existence of complement
π-bases is equivalent to the existence of Hall π-systems, and also to the existence of Sylow
π-bases, by Proposition 2.4 and the corresponding constructions. In this case the group G
satisfies Eπand Eπ,andthen,if2∈π,Gsatisfies Dπ(see [1]).
For any positive integer n,wedenotebyπ(n)the set of primes dividing n;fortheorder
|G|of a group G, we set π(G)=π(|G|).
Theorem 2.10 Assume that the group G has a complement π-basis and satisfies Dπ,where
2∈π⊆P.ThenGisπ-separable.
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Proof We argue by induction on the order of G.LetNbe a normal subgroup of G.By([18,
Theorem 7.7]), Gsatisfies Dπif and only if Nand G/Nsatisfies Dπ. We may then assume
that Gis a simple group. If |π(G)∩π|≥2, then Gwould satisfy Epand Eq, with p,qodd
different primes dividing the order of G, which would imply that Gwould not be simple by
[1, Corollary 5.5]. Consequently we may assume that |π(G)∩π|=1. On the other hand,
by hypothesis and Remark 2.9,Gsatisfies Dπand Dπ, and we may assume that the group is
neither a π-group nor a π-group. In [13, Lemma 3.1], such a simple group is characterized
to be G=PSL(2,q),whereq>3, q(q−1)≡0(3),q≡−1(4)and π(q+1)⊆π,
π(q(q−1)/2)⊆π. Hence |π(G)∩π|≥|π(q(q−1)/2)|≥2, which is not possible and
proves that Gis π-separable.
3 General framework: the theory of soluble groups
Before focusing on the universe of π-separable groups, and proving our main results in
Sect. 4, we present briefly here the general framework on the theory of soluble groups and
classes of groups, where they are relevant to our concerns. It makes clear the origin of main
concepts in our extension to π-separable groups, πa set of primes, particularly the one of
Nπ-Dnormal subgroups, associated to the class Nπof groups which are the direct product
of a π-group and a nilpotent π-group.
We recall first some basic concepts and results, which are taken from [9]:
Definitions 3.1 [9, II. Sect. 2] A class Xof groups is a formation if every epimorphic image of
a group in Xbelongs to X,andG/( N1∩N2)∈Xwhenever N1,N2Gwith G/N1,G/N2∈
X.
In this case, the X-residual of a group G, denoted GX, is the smallest normal subgroup
of Gwith quotient group in X(which exists if X=∅).
A formation Xis said to be saturated if G∈Xwhenever G/(G)∈X,where(G)
denotes the Frattini subgroup of G.
For the class X=Eπof all π-groups, πa set of primes, we set GEπ=Oπ(G)for the
Eπ-residual of the group G, also described as the subgroup generated by all π-subgroups
of G.
Definitions 3.2 [9, III. Definitions (3.2), (3.5)(b)] Let Xbe a class of groups, and Gbe a
group.
(a) A subgroup Uof Gis called an X-projector of Gif UK/Kis an X-maximal subgroup
of G/K(i.e. maximal as a subgroup of G/Kin X)forall KG. The (possibly empty)
set of X-projectors of Gwill be denoted by ProjX(G).
(b) An X-covering subgroup of Gis a subgroup Eof Gwith the property that E∈ProjX(H)
whenever E≤H≤G.ThesetofX-covering subgroups of Gwill be denoted by
CovX(G).
The classical theorem of Carter ( [7], [9, III. Theorem (4.6)]) states that each soluble group
possesses exactly one conjugacy class of Carter subgroups, i.e. self-normalizing nilpotent
subgroups, which coincide with N-projectors and with N-covering subgroups, for the class
Nof nilpotent groups.
Definitions 3.3 [9, IV. Definitions (3.1), Theorem (3.2), II. Definition (1.2)(b)]
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(a) A formation function is a function fwhich associates with each prime pa (possibly
empty) formation f(p).
(b) A class Fof groups is called a local formation if there exists a formation function fsuch
that Fconsists of the groups Gwhich satisfy that, for all chief factors H/Kof Gand for
all primes pdividing |H/K|, it holds that G/CG(H/K)∈f(p). The class Fis said to
be locally defined by the formation function f and denoted F=LF(f).
The characteristic of the local formation LF(f)is Char(LF(f)) ={p∈P|Zp∈
LF(f)}={p∈P|f(p)=∅},where Zpdenotes the cyclic group of order p.
Remark 3.4 [9, IV. Definition (3.9)(b)] A local formation Falways has a smallest local
definition, i.e. a formation function fsuch that F=LF(f),and f(p)⊆g(p)for every
prime pand any other formation function gsuch that F=LF(f)=LF(g).
The well-known theorem of Gaschütz-Lubeseder-Schmid (cf. [9, IV. Theorem (4.6)] states
that non-empty saturated formations are exactly local formations.
Let πbe a set of primes. Let
Nπ=Eπ×Nπ=(G=H×K|H∈Eπ,K∈Nπ),
Eπthe class of all π-groups and Nπthe class of all nilpotent π-groups.
In the particular cases when either π=∅or π={p},pa prime, (|π|≤1), then Nπ=N
is the class of all nilpotent groups.
Our main results Theorems 4.5 and 4.14 extend the existence and properties of Carter
subgroups in soluble groups to ρ-separable groups, ρa set of primes, with appropriate class
Nρor Nρplaying the role of the class Nof nilpotent groups.
We shall appeal also to the concept of Nπ-Dnormal subgroup, as Nπis a saturated
formation.
The concept of G-Dnormal subgroups for a non-empty saturated formation G,whichwas
given by K. Doerk in the universe of finite soluble groups, and appears for the first time in
[2, Definition 3.1], is also available for arbitrary finite groups, as defined next. For notation,
if Gis a group, ρa set of primes, Gρ∈Hallρ(G)and H≤G,wewriteGρHto mean
that Gρreduces into H, i.e. Gρ∩H∈Hallρ(H).
Definition 3.5 [2, Definition 3.1] Let Gbe a non-empty saturated formation and let Gbe a
group. A subgroup Hof Gis said to be G-Dnormal in Gif π(|G:H|)⊆Char(G),andfor
every p∈Char(G)it holds that
[Hp
G,Hg(p)]≤H,
where gdenotes the smallest local definition of Gas local formation, and Hp
G=Gp∈
Sylp(G)|GpH.
The saturated formation Nπ=Eπ×Nπ=LF(f)=LF(f)is locally defined by the for-
mation function fgiven by f(p)=Epif p∈π,and f(p)=Eπif p∈π. Then the smallest
local definition is given by f(p)=(1)if p∈π,and f(p)=(1)if π={p},
Eπif p∈π, |π|≥2.
Hence, for Nπ=N,N-Dnormal subgroups are exactly normal subgroups.
In the case |π|≥2, a subgroup Hof a group Gis Nπ-Dnormal if it satisfies the following
conditions:
(1)whenever p∈πand Gp∈Sylp(G),GpH,thenGp≤NG(H);
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(2)whenever p∈πand Gp∈Sylp(G),GpH,thenGp≤NG(Oπ(H)).
Note that normal subgroups are Nπ-Dnormal for any set of primes π.
Remark 3.6 Regarding the previous statement, note that for any X≤Git holds that
[X,Oπ(H)]≤Hif and only if X≤NG(Oπ(H)).
Proof Assume that [X,Oπ(H)]≤H.WeconsiderOπ(H)=Hq|Hq∈Sylq(H), q∈
π.Then[X,Oπ(H)]≤Oπ(H)x|x∈X=Hx
q|x∈X,Hq∈Sylq(H), q∈π=
Oπ(H). Theconverseisclear.
The next proposition provides a useful characterization of Nπ-Dnormal subgroups.
Proposition 3.7 Let H be a subgroup of a group G. Then:
1. Assume that |π|≤1.ThenNπ=Nand H is N-Dnormal in G if and only H is normal
in G.
2. Assume that |π|≥2. Then the following statements are equivalent:
(i) HisNπ-Dnormal in G;
(ii) Oπ(H)G and Oπ(G)≤NG(H).
Proof Part 1 is clear. For Part 2, since Oπ(G)=Gp|Gp∈Syl p(G), p∈π,itis
clear that (ii) implies (i). Conversely, assume the (i) holds, i.e. His Nπ-Dnormal in G.By
Sylow’s theorem, for each prime p, there exists Gp∈Syl p(G)such that GpH.The
definition of Nπ-Dnormality implies that Gp≤NG(Oπ(H)) if p∈π,andGp≤NG(H)≤
NG(Oπ(H)) if p∈π. Consequently, G=Gp|p∈P≤NG(Oπ(H)), i.e. Oπ(H)G.
In particular, for any p∈πand any Gp∈Syl p(G), it holds that Gp∩H=Gp∩Oπ(H)∈
Sylp(Oπ(H)) =Sylp(H), which means that GpH,andthenGp≤NG(H), because
His Nπ-Dnormal in G. Hence, Oπ(G)=Gp|Gp∈Sylp(G), p∈π≤NG(H),and
we are done.
For notation, whenever a group X∈Nπ,wewrite X=Xπ×Xπwhere Xπ=Oπ(X)∈
Eπand Xπ=Oπ(X)∈Nπ.
Corollary 3.8 Assume that |π|≥2and let H be a subgroup of a group G such that H =
Hπ×Hπ∈Nπ.Then HisNπ-Dnormal in G if and only if HπG and Oπ(G)≤NG(H).
Proof This is a consequence of Proposition 3.7(2) since in this case Hπ=Oπ(H).
4 Carter-like subgroups in -separable groups
Let πbe a set of primes. As above, set Nπ=Eπ×Nπ,whereEπis the class of π-groups
and Nπis the class of nilpotent π-groups.
We prove in this section that if Gis a π-soluble group, then Nπ-projectors coincide
with Nπ-covering subgroups, and they form a conjugacy class of self-Nπ-Dnormalizing
subgroups of G(see Definition 4.7, and Theorems 4.5,4.14). If Nπ=Nis the class of
nilpotent groups and Gis a soluble group, these are the Carter subgroups.
Remark 4.1 We notice that Burnside’s paqb-theorem together with the Feit-Thompson the-
orem imply that π-separable groups are π-soluble whenever |π|≤2, or 2 ∈πif |π|≥3.
Also, by the Feit-Thompson theorem, for any set of primes ρ,aρ-separable group is either
ρ-soluble or ρ-soluble.
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For our main results we quote a series of results from [9, III. Sect. 3] and adhere to
the notation there, though specialized to our saturated formation Nπand our purposes. We
notice that b(Nπ)⊆P1∪P2,whereb(Nπ)is the class of groups G/∈Nπbut whose proper
epimorphic images belong to Nπ, and for each i=1,2, Piis the class of primitive groups
with a unique minimal normal subgroup, which is abelian for i=1, and non-abelian for
i=2.
Lemma 4.2 1. [9, III. Proposition (3.7)] For a group G, whenever N G, N ≤V≤G,
U∈ProjNπ(V), and V /N∈ProjNπ(G/N),thenU ∈ProjNπ(G).
2. [9, III. Lemma (3.9)] Assume that G ∈b(Nπ). Then:
(a) If G ∈P1,thenCovNπ(G)and ProjNπ(G)both coincide with the non-empty set
comprising those subgroups of G which are complements in G to the minimal normal
subgroup of G.
(b) If G ∈P2,thenProjNπ(G)is non-empty and consists of all Nπ-maximal subgroups
of G which supplement the minimal normal subgroup of G in G.
3. [9, III. Theorem (3.10)] For any group G, CovNπ(G)⊆ProjNπ(G)=∅.
4. [9, III. Theorem (3.14)] Let N be a nilpotent normal subgroup of a group G, and let H
be an Nπ-maximal subgroup of G such that G =HN.Then H ∈ProjNπ(G).
5. ([9, III. Theorem (3.19)], [11]) Let Bbe the class of all π-soluble groups. The statement
“ProjNπ(G)is a conjugacy class of G” is true for all groups G ∈Bif and only if it is
true for all G ∈b(Nπ)∩B.
6. ([9, III. Remark (3.20)(b)], [11]) Let Bbe the class of all π-soluble groups. The statement
“ProjNπ(G)=CovNπ(G)” is true for all groups G ∈Bif and only if it is true for all
G∈b(Nπ)∩B.
We still quote the following result for our purposes.
Lemma 4.3 [5, Theorems 4.1.18, 4.2.17] Let Hbe a saturated formation and let G be a group
whose H-residual GHis abelian. Then G His complemented in G, any two complements are
conjugate in G, and the complements are the H-projectors of G.
Lemma 4.4 Let M =Mπ×Mπbe an Nπ-maximal subgroup of a π-separable group G.
Then:
1. M=MπCG(Mπ)πfor some CG(Mπ)π∈Hallπ(CG(Mπ)).
2. If H =Hπ×Hπis another Nπ-maximal subgroup of G and M x
π=Hπfor some
x∈G, then Mg=Hforsomeg∈G.
Proof 1. We have that Mπ≤CG(Mπ)and so Mπ≤CG(Mπ)πfor some CG(Mπ)π∈
Hallπ(CG(Mπ)).But MπCG(Mπ)π∈Nπ, which implies that M=MπCG(Mπ)π∈
Nπby the maximality of M.
2. The hypothesis implies that CG(Mπ)x=CG(Mx
π)=CG(Hπ).ThenMx
π,Hπ∈
Hallπ(CG(Hπ)) and Mxy
π=Hπfor some y∈CG(Hπ). Consequently, Mxy =
Mxy
πMxy
π=HπHπ=H, and we are done.
Theorem 4.5 If G is a π-soluble group, then ∅ = ProjNπ(G)=CovNπ(G)and it is a
conjugacy class of G.
Proof By Lemma 4.2, parts (3), (5), (6), we may assume that G∈b(Nπ).SinceNπis a
saturated formation, G∈P1∪P2;letNbe the minimal normal subgroup of G.
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74 Page 10 of 13 M. Arroyo-Jordá et al.
If G∈P1, the result follows by Lemmas 4.2(2)(a) and 4.3.
Assume now that G∈P2. We know by Lemma 4.2(2)(b) that ProjNπ(G)is non-empty
and consists of all Nπ-maximal subgroups of Gwhich supplement Nin G.Weprovefirst
that these subgroups are conjugate in G.
Let M=Mπ×Mπ,H=Hπ×Hπbe Nπ-maximal subgroups of Gsuch that G=
NM =NH.SinceGis π-soluble, and Nis non-abelian, Nis a π-group and, consequently,
Mπ,Hπ∈Hallπ(G). Hence, there exists x∈Gsuch that Mx
π=Hπand Mand Hare
conjugate by Lemma 4.4(2).
We claim now that ProjNπ(G)⊆CovNπ(G), which will conclude the proof.
Let M∈ProjNπ(G), i.e. M=Mπ×Mπis an Nπ-maximal subgroup of Gsuch that
G=NM.Let M≤L≤G. We aim to prove that M∈ProjNπ(L).LetT∈ProjNπ(L).We
notice that L=M(L∩N).ThenL/( L∩N)∼
=M/(M∩N)∈Nπ.SinceT(L∩N)/(L∩N)
is Nπ-maximal in L/( L∩N), it follows that L=T(L∩N). Hence Tπ,Mπ∈Hallπ(L)
and, moreover, Tand Mare Nπ-maximal subgroups of L. By Lemma 4.4(2), Tand Mare
conjugate in Land M∈ProjNπ(L).
Remark 4.6 In Theorem 4.5 the hypothesis of π-solubility cannot be weakened to π-
separability. Otherwise, for the particular case when π=∅, the result would hold for every
finite group and the formation Nπ=Nof nilpotent groups, which is not true. Particularly,
also if π=∅, one can consider for instance π=P−{2,3,5},π={2,3,5}and G=Alt(5)
the alternating group of degree 5. The group Gis obviously π-separable, the Nπ-projectors
are the N-projectors, which do not form a conjugacy class of subgroups, as they are all the
Sylow subgroups of G;andGhas no N-covering subgroups.
Definition 4.7 A subgroup Hof a group Gis said to be self-Nπ-Dnormalizing in Gif
whenever H≤K≤Gand His Nπ-Dnormal in K,thenH=K.
We prove next that Nπ-projectors are self-Nπ-Dnormalizing subgroups.
Proposition 4.8 Let H be an Nπ-projector of a π-soluble group G . Then H is self-Nπ-
Dnormalizing in G.
Proof Assume that H≤K≤Gand His Nπ-Dnormal in K. We aim to prove that H=K.
By Theorem 4.5 and Corollary 3.8,wehavethat H∈CovNπ(K)and HπK.Then
H/Hπ∈ProjNπ(K/Hπ)and H/Hπ≤KπHπ/Hπ∈Nπ,foranyKπ∈Hallπ(K),
which implies that H=KπHπ. Whence, if Kπ∈Hallπ(K),thenK=HK
πand HK
by Proposition 3.7.If H<K,then H<HKpfor some p∈πand 1 = Kp∈Syl p(K).
But HKp/H∈Nπwhich contradicts the fact that H∈CovNπ(K).
Lemma 4.9 Assume that |π|≥2,H=Hπ×Hπ,K=Kπ×Kπ∈Nπand H ≤K.Then
HisNπ-Dnormal in K if and only if HπKπ.
Proof We notice that [Kπ,Hπ]=1and[Kπ,Hπ]=1. Consequently, Corollary 3.8 implies
that His Nπ-Dnormal in Kif and only if HπKπ.
Proposition 4.10 Assume that H =Hπ×Hπ<L=Lπ×Lπ∈Nπ. Then there exists
K≤L such that H <K and H is Nπ-Dnormal in K .
Proof If |π|≤1, then Nπ=N, and the result is clear. In the case |π|≥2, by Lemma 4.9,if
Hπ=Lπ,thenHis Nπ-Dnormal in L, and we are done. Otherwise, there exists T≤Lπ
such that HπT, i.e. Hπis a proper normal subgroup of T, because Lπis nilpotent.
We can consider now the subgroup K=LπT≤Lwhich satisfies that H<Kand His
Nπ-Dnormal in K, which concludes the proof.
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Carter and Gaschütz theories beyond... Page 11 of 13 74
As a consequence we can state the following:
Corollary 4.11 If H ∈Nπis a self-Nπ-Dnormalizing subgroup of a group G, then H is
Nπ-maximal in G.
Remark 4.12 It is not true in general that Nπ-projectors of π-soluble groups are exactly
self-Nπ-Dnormalizing subgroups in Nπ. Otherwise, Nπ∩Swould be either Nor S,the
class of all soluble groups, by [3, Proposition 4.1]. But we see next that a corresponding
result to [3, Theorem 4.2] is still possible. That reference provides a corresponding result to
our next Theorem 4.14, for finite soluble groups, subgroup-closed saturated formations and
associated projectors.
As a consequence of Lemma 4.3 we can state the following.
Lemma 4.13 Let Hbe a saturated formation, X be a group and H be an H-projector of X.
Then H ∩XH≤(XH).
Theorem 4.14 For a subgroup H of a π-soluble group G the following statements are
pairwise equivalent:
1. HisanNπ-projector of G.
2. HisanNπ-covering subgroup of G.
3. H∈Nπis a self-Nπ-Dnormalizing subgroup of G and H satisfies the following prop-
erty:
If H ≤X≤G,then H ∩XNπ≤(XNπ).(*)
Proof The equivalence 1 ↔2 has been proven in Theorem 4.5. On the other hand, Proposi-
tion 4.8 and Lemma 4.13 prove 2 →3. We prove next that 3 →1.
Let H∈Nπbeaself-Nπ-Dnormalizing subgroup of Gsatisfying property (∗).Weaim
to prove that H∈ProjNπ(G).Wenoticethat His Nπ-maximal in Gby Corollary 4.11.
If G∈Nπ,thenH=Gand the result follows. So that we may assume that G/∈Nπ.We
argue by induction on the order of G.LetNbe a minimal normal subgroup of Gsuch that
N≤GNπ.
We distinguish the following cases:
Case 1. G=HN.
Case 2. HN <G.
Case 1. If Nis abelian, the result follows by Lemma 4.2(4). Assume that Nis not abelian.
Let KGsuch that G/K∈b(Nπ).ThenNis not contained in Kbecause G/N∼
=
H/(H∩N)∈Nπ. In particular, N∩K=1andN∼
=NK/Kis a minimal normal
subgroup of G/K=(NK/K)(HK/K)∈P2, with HK/K<G/Kbecause HK/K∈
Nπ. By Lemma 4.2(2)(b), HK/K≤P/Kfor some P/K∈ProjNπ(G/K).Wehave
now that H≤P<G. The inductive hypothesis implies that H∈ProjNπ(P), and from
Lemma 4.2(1), H∈ProjNπ(G), as claimed.
Case 2. In this case HN <Gand the inductive hypothesis implies that H∈ProjNπ(HN).
We prove first that HN/Nsatisfies property (∗)in G/N. Assume that HN/N≤
X/N≤G/N.If X<G,thenH∈ProjNπ(X)by inductive hypothesis and then
HN/N∩(X/N)Nπ≤((X/N)Nπ). Otherwise, X=Gand so (HN/N)∩(X/N)Nπ=
(HN/N)∩(G/N)Nπ=(HN/N)∩GNπ/N=(H∩GNπ)N/N≤(GNπ)N/N=
((G/N)Nπ)=((X/N)Nπ).
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74 Page 12 of 13 M. Arroyo-Jordá et al.
We claim that HN/N∈Nπis self-Nπ-Dnormalizing in G/N. Then the result follows
by inductive hypothesis together with Lemma 4.2(1).
Assume that HN/Nis Nπ-Dnormal in L/N≤G/N.
If L<G, the inductive hypothesis implies that His an Nπ-projector of Land so HN =L
by Proposition 4.8. So that we may assume that L=Gand HN/Nis Nπ-Dnormal in G/N.
We split the rest of the proof into the following steps:
Step 1. NG(HN)=HN.
If g∈NG(HN),then HN =HgN.SinceH∈ProjNπ(HN), and this is a conjugacy
class of subgroups of HN by Theorem 4.5,Hg=Hxfor some x∈HN. Consequently,
gx−1∈NG(H)=H,andg∈HN.
Step 2. O π(G)≤HN.
Since HN/Nis Nπ-Dnormal in G/N, we have by Proposition 3.7 and Step 1 that
Oπ(G)N/N≤NG/N(HN/N)=HN/N.
Step 3. O π(G)H=HN and it is a maximal subgroup of G.
Assume that Oπ(G)≤HN ≤T<G. The inductive hypothesis implies that H∈
ProjNπ(T). Moreover, T/Oπ(G)∈Eπ⊆Nπ. Hence T=Oπ(G)H=HN.
Step 4. G/N=(HπN/N)×Oπ(G/N)∈Nπ. In particular, GNπ=N.
We have t hat HπN/NG/Nand HπN/N∈Hallπ(G/N)by Corollary 3.8 and Step 2.
Since [Hπ,Hπ]=1andG/Nis π-separable, and equivalently π-separable, it follows by
[14, 6. Theorem 3.2] that
(HπN/N)Oπ(G/N)/Oπ(G/N)≤C(G/N)/ Oπ(G/N)(Oπ((G/N)/Oπ(G/N)))
≤Oπ((G/N)/Oπ(G/N)),
which implies HπN/N≤Oπ(G/N).
If HπN/N=Oπ(G/N),thenHN/NG/Nand G=HN by Step 1, a contradiction.
Consequently, we may assume that HπN/N<Oπ(G/N).ThenHN/N<
(HπN/N)Oπ(G/N)≤G/N.ByStep3,(HπN/N)Oπ(G/N)=G/N∈Nπ.
Step 5. G =NP where P=Pπ×Pπ∈ProjNπ(G).
It follows by Step 4.
Step 6. N ∈Eπ.
If N∈Eπ,thenHπ,Pπ∈Hallπ(G).SinceHand Pare both Nπ-maximal subgroups,
it follows by Lemma 4.4 that H=Px∈ProjNπ(G)and then HN =G, a contradiction.
Since Gis π-separable, N∈Eπ.
Step 7. Final contradiction.
Steps 4, 6, imply that N=GNπis abelian, since Gis π-soluble. By hypothesis, H∩N=
1. Moreover, PπN=HπN=Oπ(G)and Pπ∈Hallπ(G). There is no loss of generality
to assume that Hπ≤Pπand then [Hπ,Pπ]=1.
Since His an Nπ-maximal subgroup of HN =HπHπN, we deduce that Hπis an N-
maximal subgroup of CHπN(Hπ)=HπCN(Hπ).ButCN(Hπ)CHπN(Hπ)and CN(Hπ)
is nilpotent. By Lemma 4.2(4), Hπ∈ProjN(CHπN(Hπ)). On the other hand,
CHπN(Hπ)=CPπN(Hπ)=PπCN(Hπ).
Since Pπis nilpotent, it follows again by Lemma 4.2(4), that Pπ≤Hx
π∈
ProjN(CHπN(Hπ)),forsomex∈CN(Hπ).If Pπ<Hx
π,since PπN=Hx
πN,we
have that Hx
π∩N= 1 and so also H∩N= 1, a contradiction. Therefore, Pπ=Hx
π,and
Lemma 4.4 implies that Hg=Pfor some g∈Gas Hand Pare Nπ-maximal subgroups
of G. It follows that H∈ProjNπ(G)and HN =G, the final contradiction.
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Carter and Gaschütz theories beyond... Page 13 of 13 74
Acknowledgements The authors want to thank Peter Hauck for helpful conversations. This research has been
supported by Proyectos PROMETEO/2017/ 057 from the Generalitat Valenciana (Valencian Community,
Spain), and PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovación y Universidades, Spain,
and FEDER, European Union. The fourth author acknowledges with thanks the financial support of the
Universitat de València as research visitor (Programa Propi d’Ajudes a la Investigació de la Universitat de
València, Subprograma Atracció de Talent de VLC-Campus, Estades d’investigadors convidats (2019)).
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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