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mathematics
Article
Products of Finite Connected Subgroups
María Pilar Gállego 1,†, Peter Hauck 2, Lev S. Kazarin 3, Ana Martínez-Pastor 4and
María Dolores Pérez-Ramos 5,*
1Departamento de Matemáticas, Universidad de Zaragoza, Edificio Matemáticas, Ciudad Universitaria,
50009 Zaragoza, Spain; pgallego@unizar.es
2Fachbereich Informatik, Universität Tübingen, Sand 13, 72076 Tübingen, Germany;
peter.hauck@uni-tuebingen.de
3Department of Mathematics, Yaroslavl P. Demidov State University, Sovetskaya Str 14,
150014 Yaroslavl, Russia; Kazarin@uniyar.ac.ru
4Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València,
Camino de Vera, s/n, 46022 Valencia, Spain; anamarti@mat.upv.es
5Departament de Matemàtiques, Universitat de València, C/Doctor Moliner 50,
46100 Burjassot (València), Spain
*Correspondence: Dolores.Perez@uv.es
†
M. Pilar Gállego passed away on the 22 May 2019. We had the privilege to work with her and to experience
her insight and generosity to share her ideas. We miss her as a collaborator and friend.
Received: 5 August 2020; Accepted: 1 September 2020; Published: 4 September 2020
Abstract:
For a non-empty class of groups
L
, a finite group
G=AB
is said to be an
L
-connected
product of the subgroups
A
and
B
if
ha
,
bi∈L
for all
a∈A
and
b∈B
. In a previous paper,
we prove that, for such a product, when
L=S
is the class of finite soluble groups, then
[A
,
B]
is
soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose
two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite
groups previous research about finite groups in the soluble universe. In particular, we characterize
connected products for relevant classes of groups, among others, the class of metanilpotent groups
and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of
relevant subgroups of finite groups.
Keywords:
finite groups; products of subgroups; two-generated subgroups;
L
-connection; fitting
classes; fitting series; formations
MSC: 20D10; 20D40; 20D25
1. Introduction and Main Results
All of the groups considered in this paper are assumed to be finite. We extend previous research
on the influence of two-generated subgroups on the structure of groups, in connection with the study
of products of subgroups. In [1], the following result is proven:
Theorem 1.
Let the finite group
G=AB
be the product of subgroups
A
and
B
. Then the following statements
are equivalent:
1. ha
,
bi
is soluble for all
a∈A
and
b∈B
, i.e.,
A
,
B
are
S
-connected for the class
S
of all finite soluble
groups (cf. Definition 1below).
2. For all primes p 6=q, all p-elements a ∈A and all q-elements b ∈B, ha,biis soluble.
3. [A,B]≤GS, where GSdenotes the soluble radical of G (i.e., the largest soluble normal subgroup of G).
Mathematics 2020,8, 1498; doi:10.3390/math8091498 www.mdpi.com/journal/mathematics
Mathematics 2020,8, 1498 2 of 8
Obviously, for the special case
A=B=G
, the following well-known result of J. Thompson
is recovered:
Theorem 2
(Thompson, [
2
,
3
])
.
A finite group
G
is soluble if and only if every two-generated subgroup of
G
is soluble.
Thompson’s theorem has been generalized and sharpened in various ways. In particular, we
mention the extension of R. Guralnick, K. Kunyavski
˘
ı, E. Plotkin, and A. Shalev, which describes the
elements in the soluble radical GSof a finite group G.
Theorem 3
(Guralnick, Kunyavski
˘
ı, Plotkin, Shalev, [
4
])
.
Let
G
be a finite group and let
x∈G
.
Then x ∈GSif and only if the subgroup hx,yiis soluble for all y ∈G.
Again, the application of Theorem 1, with
A=G
, and
B=hxi ≤ G
, assures that
hxiGS
is a
normal (soluble) subgroup of G under the hypothesis in statement (1) and, therefore, Theorem 3is also
recovered. It is to be emphasized that we make use of this result in the proof of our Theorem 1.
This shows how an approach that involves factorized groups provides a more general setting for
local-global questions related to two-generated subgroups. A first extension of Thompson’s theorem
for products of groups was obtained by A. Carocca [
5
], who proved the solubility of
S
-connected
products of soluble subgroups. This way the following general connection property turns out to
be useful:
Definition 1
(Carocca, [
6
])
.
Let
L
be a non-empty class of groups. Subgroups
A
and
B
of a group
G
are
L
-connected if
ha
,
bi ∈ L
for all
a∈A
and
b∈B
. If
G=AB
, we say that
G
is the
L
-connected product of the
subgroups A and B.
The structure and properties of
N
-connected products, for the class
N
of finite nilpotent groups,
are well known (cf. [
7
–
9
]); for instance,
G=AB
is an
N
-connected product of
A
and
B
if and only if
G
modulo its hypercenter is a direct product of the images of
A
and
B
. Apart from the above-mentioned
results regarding
S
-connection, corresponding studies for the classes
N2
and
N A
of metanilpotent
groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [
10
,
11
];
in [
12
] connected products for the class
SπSρ
of finite soluble groups that are extensions of a normal
π
-subgroup by a
ρ
-subgroup, for arbitrary sets of primes
π
and
ρ
, are studied. The class
SπSρ
appears
in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting
formations, which comprise a variety of classes of groups, such as the class of
π
-closed soluble groups,
or groups with Sylow towers with respect to total orderings of the primes. A study in [
13
] of connected
subgroups, for the class of finite nilpotent groups of class at most 2, contributes generalizations of the
classical results on 2-Engel groups.
In the present paper, as an application of Theorem 1, we show that the main results in [
10
–
12
],
proved for soluble groups, remain valid for arbitrary finite groups. In particular, we characterize
connected products for some relevant classes of groups (see Theorem 4). For instance, we prove that for
a finite group
G=AB
, the subgroups
A
and
B
are
N2
-connected if and only if
A/F(G)
and
B/F(G)
are
N
-connected, which means that for all
a∈A
and
b∈B
,
ha
,
biN≤F(ha
,
bi)
if and only if for all
a∈A
and
b∈B
,
ha
,
biN≤F(G)
, where, for any group
X
,
F(X)
denotes the Fitting subgroup of
X
, and
XN
denotes the nilpotent residual of
X
, i.e., the smallest normal subgroup of
X
with nilpotent quotient
group. When we specialize our results to suitable factorizations, as mentioned above, we derive
descriptions of the elements in
Fk(G)
, the radical of a group
G
for the class
Nk
of soluble groups with
nilpotent length at most
k≥
1, as well as the elements in the hypercenter of
G
modulo
Fk−1(G)
, in the
spirit of the characterization of the soluble radical in Theorem 3(see Corollaries 3and 4). In particular,
this first result contributes an answer to a problem that is posed by F. Grunewald, B. Kunyavski
˘
ı and
Mathematics 2020,8, 1498 3 of 8
E. Plotkin in [
14
]. These authors present a version of Theorem 3for general classes of groups with
good hereditary properties ([14] (Theorem 5.12)), by means of the following concepts:
Definition 2
(Grunewald, Kunyavski
˘
ı, Plotkin [
14
] (Definition 5.10))
.
For a class
X
of groups and a group
G
, an element
g∈G
is called locally
X
-radical if
hghxii
belongs to
X
for every
x∈G
; and the element
g∈G
is called globally X-radical if hgGibelongs to X.
For a subset
S
and a subgroup
X
of a group
G
, we set
hSXi=hsx|s∈S
,
x∈Xi
, the smallest
X
-invariant subgroup of
G
containing
S
. For
g∈G
, we write
hgXi
for
h{g}Xi
. When
X
is a Fitting
class, the property
hghxii ∈ X
is equivalent to
g∈ hg
,
xiX
, the
X
-radical of
hg
,
xi
, as the property
hgGi ∈ X
is equivalent to
g∈GX
, and these properties are useful in the problem of characterizing
elements forming
GX
. As mentioned in [
14
] (Section 5.1), a main problem is to determine classes
X
for
which locally and globally
X
-radical elements coincide. Corollary 4gives a positive answer for the
class Nkof finite soluble groups of nilpotent length at most k≥1.
When the Fitting class
X
is, in addition, closed under extensions and contains all cyclic groups,
the condition
hghxii∈X
is equivalent to
hg
,
xi∈X
, but this is not the case for important classes of
groups, as the class
N
of finite nilpotent groups, or more generally
Nk
,
k≥
1. In this situation the
condition
hg
,
xi ∈ X
for all elements
x∈G
may well not be equivalent to
g∈GX
, but still of interest,
as shown in Corollary 3in relation with the hypercenter.
We shall adhere to the notation used in [
15
] and also refer to that book for the basic results on
classes of groups. In particular,
π(G)
denotes the set of all primes dividing the order of the group
G
.
Additionally,
A
and
Sπ
,
π
a set of primes, denote the classes of abelian groups and soluble
π
-groups,
respectively. For the class of all finite
π
-groups, the residual of any group
X
is denoted
Oπ(X)
,
and
Oπ(X)
stands for the corresponding radical of
X
. If
F
is a class of groups, then
N F
is the class of
groups that are extensions of a nilpotent normal subgroup by a group in F.
We gather next our main results. The first one extends to the universe of finite groups results for
soluble groups in [11] (Theorem 3), [10] (Theorem 1, Proposition 1) and [12] (Theorem 1).
Theorem 4. Let G =AB be a finite group, A,B≤G. Subsequently:
1. A,B are N A-connected if and only if [A,B]≤F(G).
2. A,B are N2-connected if and only if AF(G)/F(G)and BF(G)/F(G)are N-connected.
3. Let π,ρbe arbitrary sets of primes. The following are equivalent:
(i) A,B are SπSρ-connected.
(ii) π(G)⊆π∪ρ,[A,B]is soluble, [A,Oρ(B)] ∈ Sπ,[B,Oρ(A)] ∈ Sπ.
(iii) π(G)⊆π∪ρ,[A,B]∈ SπSρ,[A,Oρ(B)] ∈ Sπ,[B,Oρ(A)] ∈ Sπ.
4.
Let
F
be a formation of soluble groups containing all abelian groups. Assume that one of the following
conditions holds:
(i) One of the factors A,B is normally embedded in G (in the sense of [15] (I. Definition (7.1))).
(ii) A and B have coprime indices in G.
Then A and B are N F -connected if and only if AF(G)/F(G)and BF(G)/F(G)are F-connected.
Remark 1.
In Theorem 4(3), (ii) and (iii),
[A
,
Oρ(B)] ∈ Sπ
is equivalent to
[A
,
Oρ(B)] ≤Oπ(G)
, and also to
[A,Oρ(B)] ≤Oπ(GS)as [A,B]is soluble.
This is because
[A
,
Oρ(B)]
is subnormal in
G
, since
[A
,
Oρ(B)] hOρ(B)Ai=hOρ(B)BAi=
hOρ(B)GiG=AB.
As consequences of Theorem 4, we derive Corollaries 1–4, and point out again that corresponding
results for finite soluble groups were firstly obtained in [10] (Corollaries 1, 2, 3, 4).
Mathematics 2020,8, 1498 4 of 8
Corollary 1.
Let the group
G=AB
be the
N2
-connected product of the subgroups
A
and
B
, and let
F
be a
class of groups.
1. Assume that
(i) Fis a Q-closed Fitting class, or
(ii) Fis either a saturated formation or a formation containing N.
Afterwards, A,B∈ F implies G ∈ N F .
2. Assume that
(i) Fis a Q-closed Fitting class, or
(ii) Fis either a saturated formation or a formation of soluble groups containing N.
Then G ∈ N F implies A,B∈ N F .
As a particular case of Corollary 1, we state explicitly:
Corollary 2.
If the group
G=AB
is the
N2
-connected product of the
π
-separable subgroups
A
and
B
of
π-length at most l, πa set of primes, then G is π-separable of π-length at most l +1.
Remark 2.
Easy examples show that the bound for the
π
-length of
G
in Corollary 2is sharp. For instance,
for any
l≥
1, consider a set of primes
π6=∅
with
π06=∅
, where
π0
stands for the complement of
π
in the set
of all prime numbers, let
B
be a
π
-separable group of
π
-length
l
, such that
Oπ(B) =
1, let
p∈π
and
A
be a
faithful module for
B
over the field of
p
elements. Let
G= [A]B
be the corresponding semidirect product of
A
with B. Subsequently, A and B are N2-connected and the π-length of G is l +1.
Corollary 3.
Let
G
be a group,
g∈G
and
k≥
1. Subsequently,
hg
,
hi∈Nk
for all
h∈G
if and only if
g∈Z∞(G mod Fk−1(G)).
Remark 3.
1.
For
k=
1, Corollary 3gives a characterization of the hypercenter of a group. This particular case already
appears in [
10
] (Corollary 3) as a direct consequence of Lemma 1(2) below, and was also observed by
R. Maier, as mentioned in [14] (Remark 5.5), and referred to [16].
2.
Assume that
g∈G
, such that
hg
,
xi
is soluble for all
x∈G
. Let
l
be the highest nilpotent length of all
these subgroups, so that
hg
,
xi∈Nl
for all
x∈G
. By Corollary 3, it follows that
g∈Fl(G)≤GS
.
Accordingly, Corollary 3may be seen also as generalization of the characterization of the soluble radical in
Theorem 3.
Corollary 4. For a finite group G, an element g ∈G and k ≥1, the following statements are equivalent:
1. g ∈Fk(hg,hi)for all h ∈G, i.e., g is locally Nk-radical.
2. hg,hi ∈ N kfor all h ∈Fk(G)and hg,hi ∈ N k+1for all h ∈G.
3. g ∈Fk(G), i.e., g is globally Nk-radical.
Remark 4.
1.
P. Flavell proved the equivalence of Conditions 1 and 3 for
k=
2([
17
] (Theorem A)) and for arbitrary
k
and soluble groups ([17] (Theorem 2.1)), as also mentioned in [10] (Remark 3).
2.
For
k=
1, the Baer-Suzuki theorem states that
F(G) = {g∈G| hg
,
gxi ∈ N for every x ∈G}
. But for
k=
2, one can not conclude that
g∈F2(G)
whenever
hg
,
gxi ∈ N 2
for all
x∈G
, as pointed out by
Flavell [17].
Remark 5.
As application of Theorem 4, the hypothesis of solubility can be also omitted in Corollary 4 and
Propositions 3 and 4 of [
11
], especially in relation with saturated formations
F ⊆ N A
, such as the class of
supersoluble groups. Additionally, an extension for finite groups of Corollary 1 of [
12
], in relation with the
above-mentioned nilpotent-like Fitting formations, can be stated.
Mathematics 2020,8, 1498 5 of 8
2. Proofs of the Main Results
Lemma 1
([
9
] (Proposition 1(2),(8), Lemma 1, Proposition 3))
.G=AB
be an
N
-connected product of the
subgroups A and B. Subsequently:
1. A and B are subnormal in G.
2. A ∩B≤Z∞(G), the hypercenter of G.
3. If Fis either a saturated formation or a formation containing N, and A,B∈ F, then G ∈ F .
4.
If
F
is either a saturated formation or a formation of soluble groups containing
N
, and
G∈ F
,
then A,B∈ F.
Proposition 1
([
10
] (Proposition 1))
.
Let
F
be a formation of soluble groups containing all abelian groups.
Let G be a soluble group, such that G =AB is the N F -connected product of the subgroups A and B. Assume
that one of the following conditions holds:
1. One of the factors A,B is normally embedded in G.
2. A and B have coprime indices in G.
3. A and B are nilpotent.
Then
G/F(G) = (AF(G)/F(G))(BF(G)/F(G))
is an F-connected product of the two factors.
Proof.
Part 1 is [
10
] (Proposition 1). Parts 2 and 3 follow with the same arguments, taking into
account that, in both cases, the following fact holds: if in addition the group
G
has a unique minimal
normal subgroup
N=CG(N)
, then either
N≤A
or
N≤B
(in particular, for part 3 apply [
18
]
(Theorem 1)).
Lemma 2.
Let
G=AB
be a finite group,
A
,
B≤G
, and suppose that
[A
,
B]
is soluble. Let
a∈A
.
If BhaiGS=G, then A ≤GS.
Proof.
Because
[A
,
B]≤GS
and
BhaiGS=G
, it follows that
haiGSEG
. Therefore,
hai ≤ GS
and
BGS=G
. For any
x∈A
, we have now that
BhxiGS=G
, and so again
x∈GS
, which implies
A≤GS.
Definition 3.
We define a subset functor
T
to assign to each finite group
G
a subset
T(G)
of
G
satisfying the
following conditions:
1. T(H) = α(T(G)) for all group isomorphisms α:G−→ H.
2. T(G)∩U⊆T(U)for all groups G and all U ≤G.
Notation 1.
For any group
G
, let
P(A
,
B
,
G)
be a property on
G
,
A
,
B
, where
A
,
B
are subgroups of
G=AB
,
which satisfies the following conditions:
1. Whenever α:G−→ H is a group isomorphism, if P(A,B,G)is true, then P(α(A),α(B),H)is true.
2.
Whenever
L
,
A
,
B
are subgroups of a group
G=AB
, with
L
of the form
L= (L∩A)(L∩B) = GSXhyi
,
{X,Y}={A,B}, y ∈Y, if P(A,B,G)is true, then P(L∩A,L∩B,L)is true.
Proposition 2.
Let
S1
,
S2
be subset functors according to Definition 3,
Y ⊇ A
be a formation and
X
be a
Fitting class.
Suppose that the following statement (?)holds for all finite soluble groups.
(?)(If P(A,B,G)holds in the group G =AB,
then ha,biY≤GXfor all a ∈S1(A),b∈S2(B).
Subsequently, (?)holds for all finite groups G =AB, such that [A,B]is soluble.
Mathematics 2020,8, 1498 6 of 8
Proof.
Let
G=AB
be a finite group,
A
,
B≤G
, such that
[A
,
B]≤GS
, and assume that
P(A
,
B
,
G)
holds. Let
a∈S1(A)
,
b∈S2(B)
. We aim to prove that
ha
,
biY≤GX
. We argue by induction on
|G|
. First assume that
GShaiB=G
. Then
A≤GS
by Lemma 2. If
G=GShbi
, then
G
is soluble
and the result follows. Accordingly, we may assume that
GShbi=A(B∩GShbi)<G
. Because
b∈
S2(B)∩(B∩GShbi)⊆S2(B∩GShbi)
, and
GShbi
has the desired form in Notation 1(2), by induction
we have that
ha
,
biY≤(GShbi)X∩GS≤(GS)X≤GX
, because
ha
,
biY≤ ha
,
bi0= [hai
,
hbi]≤GS
.
If
GShaiB= (GShaiB∩A)B<G
, the same argument, with
GShaiB
playing the role of
GShbi
, proves
that
ha
,
biY≤GX
(note that
GShaiB
is a subgroup of
G
because
[A
,
B]≤GS
). The proposition
is proved.
Remark 6.
As we will see, Proposition 2provides the main tool to derive Theorem 4from Theorem 1and the
corresponding previous results in the soluble universe. In Notation 1(2), the additional restriction of subgroups
L= (L∩A)(L∩B)
to subgroups of the form
L= (L∩A)(L∩B) = GSXhyi
,
{X
,
Y}={A
,
B}
,
y∈Y
,
will be required only for the application to the proof of Part (4) of Theorem 4, as it is also the case of the following
Lemma 3. The present formulations of Notation 1and Proposition 2unify the treatment of the different parts
stated in Theorem 4.
Lemma 3.
1.
Assume that
A
is a normally embedded subgroup of a group
G=NA
where
NG
. Let
a∈A
.
Subsequently, Nhai ∩ A is normally embedded in Nhai.
2.
Assume that
A
is a normally embedded subgroup of a group
G=AB
,
B≤G
, such that
[A
,
B]
is soluble.
Subsequently, GSBhai ∩ A is normally embedded in GSBhai, for any a ∈A.
3.
Let
G=AB
a group, such that
A
and
B
are subgroups of coprime indices in
G
, and
[A
,
B]
is soluble.
Then GSBhai ∩ A and B have coprime indices in GSBhai, for any a ∈A.
Proof.
1.
Let
p∈π(A)
. We consider
a=apap0
, where
ap
,
ap0
denote the
p
-part and the
p0
-part of
a
,
respectively. Let
Mp∈Sylp(A)
, such that
ap∈Mp
. By the hypothesis, there exists
MG
, such
that Mp∈Sylp(M).
We claim that
(N∩Mp)hapi ∈ Sylp(Nhai ∩ A)
. Since
N∩AEA
, we have that
N∩Mp∈
Sylp(N∩A)
. Consequently,
(N∩Mp)hapi ∈ Sylp((N∩A)hapi)
. Because
Nhai ∩ A= (N∩
A)hai= (N∩A)hapihap0i, the claim follows easily.
We prove next that
(N∩Mp)hapi ∈ Sylp(Nhai ∩ M)
. Because
Nhai ∩ MENhai
, it will follow
that Nhai ∩ Ais normally embedded in Nhai, which will conclude the proof.
We notice that
Nhai ∩ M= (Nhap0i ∩ M)hapi
, so that it is enough to prove that
N∩Mp∈
Sylp(Nhap0i ∩ M).
Again
N∩MEM
implies that
N∩Mp∈Sylp(N∩M)
. Let
Np∈Sylp(N)
such that
N∩Mp=
Np∩Mp
. Then
N∩Mp=Np∩Mp≤Np∩M∈Sylp(N∩M)
because also
N∩MEN
.
Consequently, N∩Mp=Np∩Mp=Np∩M.
On the other hand,
Np∈Sylp(Nhap0i)
and
Nhap0i ∩ MENhap0i
, which implies that
Np∩M∈
Sylp(Nhap0i ∩ M), and we are done.
2.
Because
[A
,
B]≤GS
, we have that
BGSEG=AB =BGSA
. The result follows now from part 1.
3. Set N=BGSEG=AB, as before. Notice that
|Nhai:Nhai ∩ A|=|NhaiA:A|=|N A :A|=|G:A|.
Then gcd (|Nhai:B|,|Nhai:Nhai ∩ A|)|gcd (|G:B|,|G:A|) = 1, and the result follows.
Mathematics 2020,8, 1498 7 of 8
Proof of Theorem 4.
1.
Apply Proposition 2with
P(A
,
B
,
G)
being
A
,
BN A
-connected,
S1(G) = S2(G) = G
for all
groups G,Y=A,X=N, Theorem 1and ([11] (Theorem 3)).
2.
Apply Proposition 2with
P(A
,
B
,
G)
being
A
,
BN2
-connected,
S1(G) = S2(G) = G
for all groups
G,Y=N,X=N, Theorem 1and [10] (Theorem 1).
3. (i) =⇒(ii):
Apply Proposition 2with
P(A
,
B
,
G)
being
A
,
BSπSρ
-connected,
S1(G) = G
,
S2(G) = {g∈
G|gis a ρ0-element}, for all groups G,Y=A,X=Sπ, Theorem 1and [12] (Theorem 1).
(ii) =⇒(iii):
With the notation of Proposition 2, let P(A,B,G)be defined, as follows:
•π(G)⊆π∪ρ,[A,Oρ(B)] ∈ Sπ,[B,Oρ(A)] ∈ Sπ.
In addition set S1(G) = S2(G) = G, for all groups G,Y=A,X=SπSρ.
We notice that, in this case, the condition
ha
,
biY=ha
,
bi0= [hai
,
hbi]≤GX=GSπSρ
for all
a∈Aand b∈B, is equivalent to [A,B]∈ SπSρ.
We prove next that whenever
G=AB ∈ S
,
π(G)⊆π∪ρ
,
A
,
B≤G
,
[A
,
Oρ(B)] ∈ Sπ
,
[B,Oρ(A)] ∈ Sπ, then [A,B]∈ SπSρ.
For such a group G=AB, we argue as in the proof of [12](Theorem 1, (b)⇒(a)) and consider
A=Oρ(A)Aρ
and
B=Oρ(B)Bρ
, where
Aρ
and
Bρ
are Hall
ρ
-subgroups of
A
and
B
,
respectively, such that AρBρis a Hall ρ-subgroup of G. Subsequently:
[A,B]Oπ(G) = [A,Oρ(B)Bρ]Oπ(G) = [A,Bρ]Oπ(G) = [ Aρ,Bρ]Oπ(G).
Because AρBρ∈ Sρ, it follows that [A,B]∈ SπSρ.
We can apply now Proposition 2to deduce that (ii) implies (iii).
(iii) =⇒(i):
Apply Proposition 2with P(A,B,G)being:
•π(G)⊆π∪ρ,[A,Oρ(B)] ∈ Sπ,[B,Oρ(A)] ∈ Sπ,
S1(G) = S2(G) = G, for all groups G,Y=SπSρS(π∪ρ)0,X= (1), and [12] (Theorem 1).
4.
Apply Proposition 2with
P(A
,
B
,
G)
being
A
,
BN F
-connected, either
A
or
B
normally embedded
in
G
for the case (i), or
A
and
B
of coprime indices in
G
for the case (ii),
S1(G) = S2(G) = G
for
all groups G,Y=F,X=N, Lemma 3, Theorem 1and Proposition 1.
Proof of Corollary 1.
If
G=AB
is an
N2
-connected product of subgroups
A
and
B
, then
G/F(G)
is
the
N
-connected product of the subgroups
AF(G)/F(G)
and
BF(G)/F(G)
by Theorem 4(2). The result
follows now from Lemma 1.
Proof of Corollary 3.Mimic the proof of [10] (Corollary 3) by applying Theorem 4(4)(i).
Proof of Corollary 4.Mimic the proof of [10] (Corollary 4) by using now Corollary 3.
Author Contributions:
M.P.G., P.H., L.S.K., A.M.-P. and M.D.P.-R. have contributed equally in the tasks of
conceptualization, methodology, validation, investigation and writing—original draft preparation. All authors
have read and agreed to the published version of the manuscript.
Funding:
Research 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; and third author also by Project VIP-008 of Yaroslavl P. Demidov State University.
Conflicts of Interest: The authors declare no conflict of interest.
Mathematics 2020,8, 1498 8 of 8
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