Generalized norms of groups
In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subg...
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nasplib_isofts_kiev_ua-123456789-1557452025-02-23T19:49:08Z Generalized norms of groups Drushlyak, M.G. Lukashova, T.D. Lyman, F.M. In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups. 2016 Article Generalized norms of groups / M.G. Drushlyak, T.D. Lukashova, F.M. Lyman // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 48-81. — Бібліогр.: 127 назв. — англ. 1726-3255 2010 MSC:20D25, 20E28 https://nasplib.isofts.kiev.ua/handle/123456789/155745 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups. |
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Drushlyak, M.G. Lukashova, T.D. Lyman, F.M. |
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Drushlyak, M.G. Lukashova, T.D. Lyman, F.M. Generalized norms of groups Algebra and Discrete Mathematics |
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Drushlyak, M.G. Lukashova, T.D. Lyman, F.M. |
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Generalized norms of groups |
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Generalized norms of groups |
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Generalized norms of groups |
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Generalized norms of groups |
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Generalized norms of groups |
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generalized norms of groups |
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Інститут прикладної математики і механіки НАН України |
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Generalized norms of groups / M.G. Drushlyak, T.D. Lukashova, F.M. Lyman // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 48-81. — Бібліогр.: 127 назв. — англ. |
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Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics SURVEY ARTICLE
Volume 22 (2016). Number 1, pp. 48–81
© Journal “Algebra and Discrete Mathematics”
Generalized norms of groups
Maryna G. Drushlyak, Tetyana D. Lukashova,
Fedir M. Lyman
Communicated by I. Ya. Subbotin
Abstract. In this survey paper the authors specify all
the known findings related to the norms of the group and their
generalizations. Special attention is paid to the analysis of their
own study of different generalized norms, particularly the norm of
non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the
norm of infinite subgroups, the norm of infinite Abelian subgroups
and the norm of other systems of Abelian subgroups.
Introduction
In group theory findings related to the study of characteristic sub-
groups (in particular, the center, the derived subgroup, Frattini subgroup,
etc.) and the impact of properties of these subgroups on the structure
of the group are in the focus. Nowadays the list of such characteristic
subgroups can be broaden by means of different Σ-norms of a group.
Let Σ be the system of all subgroups of the group which have some
theoretical group property. For example, Σ can consist of all subgroups
of the group, of all cyclic, all non-cyclic, all Abelian, all non-Abelian,
all subnormal, all maximal, all infinite subgroups of the group. The
intersection NΣ(G) of the normalizers of all subgroups of the group which
2010 MSC: 20D25, 20E28.
Key words and phrases: norm of group, generalized norm, Wielandt subgroup,
norm of non-cyclic subgroups, norm of Abelian non-cyclic subgroups, norm of infinite
subgroups, norm of infinite Abelian subgroups, norm of decomposable subgroups.
M. Drushlyak, T. Lukashova, F. Lyman 49
belong to the system Σ is called Σ-norm of a group G. In the case Σ=∅
we assume that G=NΣ(G).
By the definition of the Σ-norm it follows that it is a characteristic
subgroup of the group and contains the center of the group. Also, NΣ(G)
is the maximal subgroup of the group that normalizes all Σ-subgroups
of the group. Therefore, all subgroups of the Σ-norm, which belong to
the system Σ, are normal in NΣ(G) (although these subgroups may not
exist).
Considering the Σ-norm, there are several problems related to the
study of the group properties with the given system Σ of subgroups and
some restrictions, which the norm satisfies. Many algebraists solved the
similar problems but the choice of a system Σ and properties of the
Σ-norm varied.
Knowing the structure of Σ-norms and the nature of its attachment
to the group, the properties of the group can be characterized in many
cases. For example, if the Σ-norm coincides with the group and Σ 6= ∅,
then all subgroups of the system Σ are normal in the group. First non-
Abelian groups with this property were considered in the XIX century by
R. Dedekind [1], who gave a complete description of finite non-Abelian
groups, all subgroups of which are normal, and called them Hamiltonian
groups. Infinite Hamiltonian groups were described in 1933 by R. Baer
[2]. Sets of Abelian and Hamiltonian groups combined are called the set
of Dedekind groups.
However, the study of groups with other systems Σ of normal subgroups
were continued only in the second part of the XX century, that slowed
down the study of Σ-norms. The findings of S. M. Chernikov and his
disciples are from the very field of the research. Thus nowadays the
structure of groups that coinside with the norm NΣ(G) is known for many
systems of subgroup. So the question on the study of the properties of
groups, in which the Σ-norm is a proper subgroup, arises naturally.
1. The norm of group and subgroups close to it
For the first time the problem of the study of the properties of groups,
which differ from the Σ-norm, was formulated by R. Baer in 30s of the
previous century. In [3] he introduced a subgroup N(G), which is the
intersection of normalizers of all subgroups of a group, and called it the
norm of the group G. It is clear that the norm N(G) is the Σ-norm of the
group for the system Σ, which consists of all subgroups of the group. The
50 Generalized norms of groups
norm N(G) is contained in all other Σ-norms and they can be considered
as its generalizations. It is also clear that Dedekind groups coincide with
their norms, so the index of the norm in a group can serve as a certain
"degree of Dedekindness” of a group.
The norm of a group was studied by R. Baer [3–10] and several other
authors [11–28]. R. Baer noticed that the restrictions that are imposed
on the norm of the group influence the structure of the group in a certain
way. Thus, there is a proposition.
Proposition 1.1 ([3]). If a norm N(G) of a group G is Hamiltonian,
then the following propositions take place:
1) G is a periodic group;
2) G contains no elements which orders are divisible by 8;
3) all elements of a group G, which have orders multiple of 4, can be
represented in the form of za, where a ∈ N(G), |a| = 4, and z ∈ G,
more over z is permutable with each element of a norm N(G);
4) any element, which order is not divisible by 4, is permutable with
every element of the subgroup N(G).
Studying the relations between the norm and the center of the group
R. Baer showed that the norm coincides with the center of the group, if
it contains elements of infinite order [3]. Another important result, which
specifies the influence of the center of the group on its norm, was offered
in [10].
Proposition 1.2 ([10]). The norm of a group G is identity if and only
if G is a group with an identity center.
Developing the study of the properties of the norm of a group L. Wos
[11] found out that the norm N(G) is contained in the third hypercentre
of the group, and the group of automorphisms, which are induced on the
subgroup N(G) by G, is nilpotent of class at most 2. In addition, it was
proved that the norm of the group is contained in the second hypercentre
if and only if the group of automorphisms induced on N(G) by the group
G is Abelian. This result was substantially refined by E. Shenkman in
[12].
Proposition 1.3 ([12]). The norm N(G) of a group G is contained in
the second hypercentre of G. The derived subgroup G′ is a subgroup of
centralizer of a norm N(G) in G.
M. Drushlyak, T. Lukashova, F. Lyman 51
So the group of automorphisms induced on N(G) by G is Abelian. Let’s
note that N(G) = E in groups with the identity center by Proposition 1.3.
In [6,8] the properties of periodic groups with an Abelian norm quotient
group were considered. In particular, in [6] it was proved that a periodic
group G, which quotient group G/N(G) is Abelian and N(G) 6= Z(G),
is a direct product of its primary components, and its norm N(G) is a
direct product of norms of these components.
In this regard, let’s note that unlike some other characteristic sub-
groups (the center of the group, the derived subgroup, Fitting subgroup
and others) the norm of the direct product of arbitrary subgroups is not
equal to the direct product of the norms of the correlative components in
general case.
Example 1.1. Let G = Q × B, where B is a non-periodic Abelian group
of rank 1, Q is a quaternion group of order 8. In this group
N(G) = N(Q × B) = Q2 × B 6= N(Q) × N(B) = G.
The problem of finding the norms of direct products of groups was
studied by J. Evan in [13].
Let’s also regard the following finding of R. Baer, which characterizes
the properties of p-groups with an Abelian norm quotient group.
Proposition 1.4 ([8]). If G is a p-group (p 6= 2, p 6= 3) that has an
Abelian quotient group for the norm N(G), more over N(G) 6= Z(G) and
pr is the exponent of the group CG(N(G)), then:
1) G is a group of finite exponent;
2) N(G)/Z(G) is a cyclic group, the order of which is equal to the
exponent of the group of automorphisms induced on N(G) by G;
3) centralizer CG(N(G)) consists of those and only those elements
x ∈ G for which xpr
= 1.
The restrictions p 6= 2, p 6= 3 in Proposition 1.4 are significant, as it is
illustrated by the examples of the respective groups (see [6]).
Nowadays the interest to the norm N(G) of a group is not reduced,
as research works [13–31], devoted to the study of its properties, are still
numerous. Thus, in [17,18] R. Bryce and J. Cossey considered series of
norms
1 = N0(G) ⊆ N1(G) ⊆ . . . ⊆ Ni(G) ⊆ . . . ,
where Ni(G)/Ni−1(G) = N(G/Ni−1(G)) for i > 1.
52 Generalized norms of groups
It was proved that in the class of 2-groups from the fact that quotient
group Ni+1(G)/Ni(G) is Hamiltonian it follows that Ni+1(G) = G. More-
over a finite 2-group, in which the quotient group G/N(G) is Hamiltonian,
but any quotient group Ni(G)/Ni−1(G) is not Hamiltonian, has order 27
and is uniquely determined up to isomorphism [18].
Starting from R. Baer, L. Wos and E. Schenkman studies of the norm
N(G) focus on its relation to the centre of the group. In particular, in
[19] J. Beidleman, H. Heineken and M. Newell have shown that in an
arbitrary p-group G either quotient group G/Z(G) or group [G, N(G)] is
cyclic. In this article the problem of the influence of properties of a norm
of a group and its center on the capability of a group G is considered.
A group G is called capable, if it is a group of inner automorphisms
of some group H that is G ∼= H/Z(H). R. Baer [29] studied such groups
for the first time. He described capable finitely generated Abelian groups
G = Zn1
⊕ Zn2
⊕ . . . ⊕ Znk
,
where ni+1
...ni, ni ∈ N
⋃
{0} and Zni
= Z is an infinite cyclic group for
ni = 0. It was found out that the group G is capable if and only if k > 2
and nk−1 = nk. The Baer’s characterization remains the only complete
one for a certain class of capable groups today.
Developing studies of the norm of a group in capable groups, X. Guo
and X. Zhang [20] in 2012 established necessary and sufficient conditions
for the coinsiding of the norm of the group with its centre, and also
dwelled upon the properties of the norm N(G) in the class of nilpotent
groups with a cyclic derived subgroup.
In 2005 N. Gavioli, L. Legarreta, S. Sica, M. Tota [22] considered
the relations between the centre Z(G), the norm N(G) and the second
hypercentre Z2(G) depending on the number v(G) of conjugacy classes
of non-normal subgroups and the number w(G) of conjugacy classes of
subgroups, which are normalizers of some subgroups, in finite p-groups
(p 6= 2) of nilpotency class c.
In 2008 F. Russo [23] studied the relations between the centre Z(G),
the norm N(G), the quazicenter Q(G) and the hyperquazicenter Q∗(G)
of quazicentral-by-finite groups. Let’s regard that the quazicenter Q(G)
of a group G is the subgroup, generated by all elements x of a group
G, such that the subgroup 〈x〉 is permutable in a group G (with other
subgroups). Accordingly, the hyperquazicenter Q∗(G) of a group G is the
M. Drushlyak, T. Lukashova, F. Lyman 53
largest term of the chain of normal subgroups
E = Q0(G) 6 Q1(G) = Q(G) 6 . . . 6 Qα 6 Qα+1 6 . . . ,
where α is an ordinal and Qα+1(G)/Qα(G) = Q(G/Qα(G)).
Proposition 1.5 ([23], Proposition 3.2). Let G be a quasicentral-by-finite
group, Q(G) be the quasicenter of G, N(G) be the norm of G, then
1) if Q(G) contains only elements of prime or infinite order and
Q(G) = N ′, where N is the subgroup generated by the quasicentral
elements of infinite order, then G is finite;
2) if there is an element x ∈ N(G) such that the index |Q(G) : 〈x〉G|
is finite, then G is central-by-finite;
3) G is central-by-finite if and only if the index |Q(G) : N(G)| is finite.
The relations between the norm N(G) and the center Z(G) in the
class of finite groups have also been studied by I. V. Lemeshev in [24].
His findings add much to Baer’s results related to finite groups.
The study of finite groups, in which Baer norm has a certain index, is
very effective. In particular, in [25] J. Wang and X. Guo studied finite
p-groups, in which the norm has a prime index, in [26] they studied finite
groups, in which the norm is a subgroup of index p or pq, where p and q
are different prime numbers. J. Smith [27] studied groups in which each
subgroup of the norm is normal in the group.
Subgroups of an arbitrary group can be considered as elements of some
subgroups lattice L(G) relative to the operations of union and intersection,
ordered by inclusion. In this sense, the norm N(G) of a group can be
defined as following [28]:
N(G) =
⋂
X∈L(G)
NG(X).
In this context, in [28] the relation between the non-cyclicness of the
norm N(G) on the one hand and the subgroup lattice L(G) of the group
G and generalized degree of commutativity of the group G on the other
hand is under the analysis.
A question naturally arises why this characteristic subgroup, in con-
trast to the center and the derived subgroup, did not get adequate attention
in the early development of group theory in view of the simple definition
of the norm and its usefullness in the study of groups. G. Miller [31]
explains that at that time other problems were posed in algebra and the
54 Generalized norms of groups
main focus of group theory has been directed to the study of solutions of
algebraic equations (in this theory simple groups play a fundamental role,
while the norm of a simple group of composite order is identity). The
norm was also not of high importance in the study of permutation groups
of low degrees, which were used in the theory of algebraic equations at
that time. The smallest degree of permutation group, which has the norm
of prime index, is equal to 8, moreover only one of 200 groups of this
order has the norm of prime index. And perhaps R.Baer drew attention
to this characteristic subgroup only in 1934 for these reasons.
Considering the intersection of normalizers of subgroups of the group,
we can get subgroups associated with Baer norm. These are the intersection
of normalizers of all subgroups contained in the given subgroup [32–
34] or conversely the intersection of the normalizers which contain the
given subgroup [35]. In particular, the concept of invariator IG(A) of the
subgroup A in the group G, which was introduced by I. Ya. Subbotin, is
the closest to the concept of the norm N(G) of the group G.
Invariator IG(A) of subgroup A in the group G [32] (quazicenralizer
[34]) is the intersection of normalizers of all subgroups of the group A in
G. This subgroup can also be called the norm of the subgroup A in the
group G [36]. In the case when the subgroup A coincides with the whole
group G the invariator IG(G) is exactly the norm N(G) of the group G.
In 2001 M. De Falco, F. de Giovanni, C. Musella [35] introduced the
concept of H-norm of the group G for some subgroup H of the group
G. H-norm of a group G is called a subgroup ker(G : H) that consists
of all elements which normalize every subgroup of X in G containing H.
Obviously, if H 6 K 6 G, then H 6 ker(G : H) 6 NG(H), ker(G : H) 6
ker(G : K). Let’s note that the E-norm, where E is the identity subgroup
of a group G, coincides with the norm N(G) of the group G.
It is clear that the norm N(G) can be defined as the subgroup of a
group G consisting of all elements of this group, which normalize every
subgroup in G. Replacing the condition of normality to pronormality we
get some analogue of the norm of a group for pronormality. It is called
pronorm P (G).
Let’s regard that an element x of the group G pronormalizes subgroup
H of a group G, if subgroups H and Hx are conjugate in 〈H, Hx〉. Ac-
cordingly, the pronorm P (G) of a group G is the set of all elements of a
group G which pronormalize every subgroup of a group. For the first time
the concept of pronorm P (G) group was introduced by F. de Giovanni,
S. Vincenzi [37] in 2000.
M. Drushlyak, T. Lukashova, F. Lyman 55
In contrast to the norm of a group the pronorm is not always a
subgroup of a group. In [38] some classes of groups, in which the set of
all elements of a group G that pronormalize every subgroup of a group,
forms a subgroup, were studied.
Proposition 1.6 ([37]). If G is polycyclic group, then its pronorm P (G)
is a subgroup.
In this work a similar statement for the class of locally soluble groups
was proved.
Subgroups generated by normalizers of given subgroups are considered
in some researches about groups with restrictions on normalizers of given
systems of subgroups. In this context, let’s consider the research of J.
Smith [39], who studied the subgroup R = R(G) generated by all proper
normalizers, and called it conorm of a group. If the group G has not
proper normalizers, then the group G is Dedekind and R(G) = E.
In 1990 H. Bell, F. Guzman, L.-Ch. Kappe [40] studied so-called Baer-
kernel, which is a ring analogue of the norm of the group. Baer-kernel of
the ring K is defined as the set
B(K) = {a ∈ K|∀y ∈ K, ∃r, s ∈ N(ay = yra ∧ ya = ays)}.
In 2010 year M. R. Dixon, L. A. Kurdachenko, D. Otal used the so-
called norm of subspace in linear groups in the research of linear groups
with finite dimensional orbits [41].
Let A be a vector space over a field F , GL(F, A) be a group of all
automorphisms of a space A, G be a subgroup of a group GL(F, A), B
be a subspace of a space A. The norm of the subspace B in the group G
is the intersection of normalizers of all F -subspases in B:
NormG(B) =
⋂
b∈B
NG(bF ).
It is known when the group G coincides with the norm NormG(B),
then the group G is isomorphic to a subgroup of the multiplicative
group U(F ). If the group G has finite dimensional orbits over A, then A
contains a FG-submodule D of finite dimension dimF (D). If K = CG(D),
then K 6 NormG(A/D). When G-orbits of every subspace from A are
finite, then A contains a FG-submodule B such that dimF (A/B) and
|G : NormG(B)| are also finite.
56 Generalized norms of groups
Therefore, the research devoted to the study of the norms of the group
and related subgroups is a very important and interesting direction in the
group theory. At the same time, there are still many questions regarding
the structural characteristics of the group depending on the structure
of its norms, conditions of coinsiding of the norm of the group and its
center, etc. left.
2. Generalized norms of some systems of maximal and
subnormal subgroups
As noted above, the norm N(G) is the Σ-norm of the group, in which
the system Σ is a system of all subgroups of this group. Narrowing the
system of all subgroups, for example, to the system of all Abelian or all
maximal subgroups of a group, we will get new Σ-norms, which can be
considered as generalizations of the norm N(G).
The first generalizations of this kind were made in the 50-th of the XX
century. In particular, in 1953 R. Baer [42] considered the intersection
H(G) of normalizers of all Sylow subgroups of a group G and called this
intersection as hypercenter of a group G. It is clear that hypercenter H(G)
is the Σ-norm, where the system Σ consists of all Sylow subgroups of
the group. R. Baer proved that H(G) coincides with the intersection of
all maximal nilpotent subgroups, and the quotient group G/H(G) is a
group with an identity center. Moreover, it was found out that the normal
subgroup belongs to a hypercenter if and only if its elements of order pn
generate cyclic subgroups of index pn.
In 1968 B. Huppert [43] generalized the concept of a hypercenter
introducing the concept of ℑ-hypercenter. Let ℑ be a class of finite groups
which can be represented as direct products of their Hall π-subgroups
with respect to some partition of non-empty set π of all primes. This class
is a local formation. The chief factor H/K of a group G is called ℑ-central
[44], if H/Kλ(G/CG(H/K)) ∈ ℑ. The product of all normal subgroups of
G which G-chief factors are ℑ-central in G is called ℑ-hypercenter Zℑ(G)
of a group G [45]. In 2013 V. I. Murashka [46] studied the properties of
ℑ-hypercenter and got some Baer’s results on the norm of the group as
corollaries in some cases.
One of the mentioned generalizations of the norm of the group is a so-
called A-norm NA(G) of the group G. It is the intersection of normalizers of
all maximal Abelian subgroups. This norm was introduced by W. Kappe
[47] in 1961. As it turned out (see [47]) in finite group A-norm is a
M. Drushlyak, T. Lukashova, F. Lyman 57
subgroup, each element of which is permutable with its conjugate (such
groups were studied, in particular, by F. Levi [48]). In addition, it was
found that the A-norm is close to a subgroup of right Engel elements of
length 2, that allowed to use it in the study of Engel groups.
Let’s regard (see e.g. [49]) that the element x ∈ G is called the right
Engel element of length 2, if for any element g ∈ G there is a relation
[[x, g], g] = 1.
Let R(G) denote the subgroup of a group G generated by all right
Engel elements of length 2 of a group G. The following propositions take
place.
Proposition 2.1 ([47]). A-norm NA(G) of a group G contains the second
hypercenter of a group G and is contained in the subgroup R(G). Moreover
the quotient group R(G)/NA(G) is elementary Abelian group of exponent
not exeeding 2.
Proposition 2.2 ([47]). For an element x ∈ G which order is not divisible
by 2, the following statements are equivalent:
1) x ∈ NA(G);
2) x is right Engel element of length 2 in G;
3) if 〈x〉 ⊳ G and U is the group of automorphisms induced on 〈x〉 by
G, then x belongs to A-norm of the group 〈x〉U ;
4) for any elements g, h ∈ G the equality [[x, g], h] = [[x, h], g]−1 takes
place.
The following proposition on a A-norm is a generalization of Wos’ [11]
and Schenkman’s results [12] related to the norm N(G) of the group.
Proposition 2.3 ([47]). Group G induces on the subgroup NA(G) a
nilpotent group of automorphisms. Its class of nilpotency does not exceed 2.
Later on W. Kappe [50–52] generalized the concept of the A-norm of
the group and introduced a so-called E-norm, which was defined as the
intersection of normalizers of all maximal subgroups of the group with
the given theoretical group property E. Clearly, E-norm NE(G) contains
the norm N(G). The intersection of an arbitrary subgroup of a group G
and the E-norm of the group is contained in the E-norm of this subgroup.
Besides NE(NE(G)) = NE(G).
A subgroup ∆(G) is related to the concept of the E-norm. It was
studied by W. Gashutz [53] and was defined as the intersection of nor-
malizers of all maximal subgroups of the group. It is clear that Gashutz
58 Generalized norms of groups
subgroup ∆(G) can be considered as Σ-norm of a group for the system Σ
that consists of non-normal in G maximal subgroups. In [53] it was found
out that ∆(G) is nilpotent and ∆(G)/Φ(G) = Z(G/Φ(G)), where Φ(G)
is Frattini subgroup.
In 1958 H. Wielandt [54] studied the properties of normalizers of
subnormal subgroups and introduced the subgroup W (G). It is the inter-
section of normalizers of all subnormal subgroups of a group. It is clear
that Wielandt subgroup W (G) is the norm of subnormal subgroups of a
group.
It is obvious that a subnormal norm coincides with the norm N(G) in
a nilpotent group. In addition, the condition G = W (G) is equivalent to
the fact that all subnormal subgroups of a group are normal. By Theorem
13.3.7 [55] Wielandt subgroup W (G) contains every simple non-Abelian
subnormal subgroup of G and every minimal normal subgroup of G which
satisfies the minimal condition for subnormal subgroups. Therefore, the
subgroup W (G) is not identity in a finite group G [54].
D. Robinson [56] and J. Roseblade [57] independently from each other
got similar results for some classes of infinite groups.
Proposition 2.4 ([56,57]). If a group G satises the minimal condition
for subnormal subgroups, then the quotient group G/W (G) is finite.
These results were summarized by J. Cossey [58] for polycyclic
groups. It was found out that these groups have a finite quotient group
G/CG(W (G)).
Wieland subgroup and its generalizations were studied intensively
by O. Kegel [59], J. Cossey, R. Bryce [60–62], R. Brandl, F. Giovanni,
S. Franciosi [63]. A. Camina [64], C. Casolo [65, 66], E. Ormerod [67],
C. Wetherell [68, 69], X. Zhang and X. Guo [70,71].
In [60] it was proved that the subnormal norm W (G) is contained
in the FC-centre in a finitely generated soluble-by-finite group of a
finite rank. Furthermore, if the norm W (G) coincides with the whole
group, then all subnormal subgroups are normal in this group, that is,
the normality is transitive relation. Groups with such a property were
studied by D. Robinson in [72] and were called T -groups. If G is a finite
soluble T -group and G/L is the unique maximal nilpotent quotient group
of group G, then the quotient group G/L is Abelian or Hamiltonian and
L is Abelian.
In 1989 J. Cossey, R. Bryce [60] introduced local Wielandt subgroup
W p(G) that is the intersection of normalizers of all p′-perfect subnormal
M. Drushlyak, T. Lukashova, F. Lyman 59
subgroup of a group G. Let’s regard that the p′-perfect group is a group
that has no non-identity quotient groups of order coprime with p.
In 1992 C. Casolo [66] studied a special subgroup of a group W (G),
which was called strong Wielandt subgroup W (G), and defined as the
intersection of the centralizers of nilpotent subnormal quotient groups of
the group G:
W (G) =
{
g ∈ G|[S, g] 6 SR for all S << G
}
,
where SR is nilpotent residual of the subgroup S or the smallest normal
subgroup N of S such that the quotient group S/N is nilpotent. C. Casolo
proved that strong Wielandt subgroup W (G) is non-identity in a finite
group. Note that this subgroup was also studied by C. Wetherell [68, 69].
In 1990 R. Bryce [62] introduced one more generalization of Wielandt
subgroup, so-called m-Wielandt subgroup Um(G) of a group G that is the
intersection of normalizers of all subnormal subgroups of a group G with
a defect at most m for integer m > 1. He studied a polynilpotent lattice
of finite soluble groups in terms of Wielandt m-length. The concept of
m-series of Wielandt group is widely used. It is defined as following: for
each natural m > 1, Um,0(G) = E; if i > 1, then Um,i(G) is determined
from the condition
Um,i(G)/Um,i−1(G) = Um(G/Um,i−1(G)).
If Um,n(G) = G for some integer n, then such a minimal number n
is called Wielandt m-length. R. Bryce proved that there are limits of
commutator length and Fitting length of finite soluble groups in terms
of Wielandt m-length (m > 2), and identified the best such a restriction.
Properties of Wielandt m-subgroup Um(G) have also been studied by
C. Franchi [76,77].
In 1995 J. Biedleman, M. Dixon, D. Robinson [73,74] considered one
more Σ-norm of a group – generalized Wielandt subgroup IW (G) which
is the intersection of normalizers of all infinite subnormal subgroups of a
group. It is clear that IW (G) is a characteristic subgroup and contains
a subnormal norm W (G). If G = IW (G), then all infinite subnormal
subgroups are normal in the group. Such groups have been studied by
F. Giovanni, S. Franciosi [75] and were called IT -groups. In [73] the
structure of the group G with the property IW (G) 6= W (G) and the
structure of the quotient group IW (G)/W (G) were studied.
60 Generalized norms of groups
In [78] F. Mari, F. Giovanni introduced a new Σ-norm, in which
system Σ consists of all nonsubnormal subgroups of a group. This norm
of nonsubnormal subgroups was denoted by W ∗(G). It is clear that if
W ∗(G) = G, then all subgroups are subnormal in a group. Moreover, if
a group G is a group with a finite number of normalizers of subnormal
subgroups, then the quotient group G/W ∗(G) is finite [78].
Let’s also mention the research [79], in which so-called generalized
N -Wielandt subgroup WN (G) was introduced. It consists of all elements
of the group G, which normalize all subnormal subgroups of N . It is a
normal subgroup and, in general, may differ from N .
It is clear that W (G) ⊆ WN (G), in particular, W (G) = WN (G), if
N = G, or N = W (G), or N is the unique maximal normal subgroup. If
G is a T -group and N is a normal subgroup of G, then WN (G) = G. The
following example proves that the converse is not true.
Example 2.1 ([79]). Let G = D8 = 〈x, y〉, x8 = y2 = (xy)2 = 1,
N1 = 〈x2〉, N2 = 〈x〉, then WN1
(G) = WN2
(G) = G, but G is not a
T -group.
3. Generalized norms of characteristic subgroups
of a group
Nowadays algebraists direct their attention to a generalization of the
norm when the system Σ is selected as a system of some characteristic
subgroups. In this context Sh. Lia and Zh. Shen [80,81] considered the
Σ-norm D(G) of a finite group, where the system Σ is chosen as a system
of derived subgroups of all subgroups of the group. The authors proved
that in the case when D(G) contains all the elements of prime order,
the group G is solvable of Fitting length at most 3. In the case when
G = D(G), derived subgroup G′ is nilpotent and G′′ has nilpotency class
at most 2.
Recently a number of researches concern the norms of different systems
of residuals. In particular, Zh. Shen, W. Shi and G. Qian [82] studied the
norm S(G) of nilpotent residuals of all subgroups of prime order. It was
proved that if all elements of prime order of a finite group G are contained
in the norm S(G), then the group G is solvable. L. Gong and X. Guo
[83] studied the norm of nilpotent residuals of all subgroups of a finite
group. N. Su and Ya. Wang [84] considered the norm DF(G) of F-residual
GF of all subgroups of the group G and the norm DF
p (G)HFOp′(G) of all
M. Drushlyak, T. Lukashova, F. Lyman 61
subgroups H of a finite group G, where F is the formation. Recall that
F-residual GF of a group G is the smallest normal subgroup N of G such
that G/N ∈ F.
X. Chen and W. Guo [85] introduced the hF-norm NhF(G) of a group G.
It is the intersection of normalizers of products of F-residuals of all
subgroups of a group G and h-radical of a group G
Nh,F(G) =
⋂
H6G
NG(HFGh),
where h is Fitting class, F is formation. Let’s regard that h-radical Gh of
a group G is maximal normal h-subgroup of a group G.
If h = 1, then the subgroup N1,F(G) is called F-norm NF(G) of a
group G and defined as
NF(G) =
⋂
H6G
NG(HF).
If h = Gπ, where Gπ is the class of finite π-solvable groups, then the
subgroup NGπ ,F(G) is called πF-norm NπF(G) of a group G and defined
as
NπF(G) =
⋂
H6G
NG(HFOπ(G)).
X. Chen and W. Guo studied the properties of hF-norm, in particular,
πF-norm of a finite group G and the relations between π′F-norm and
πF-hypercentre of a group G.
In 2014 A. Ballester-Bolinches, J. Cossey, L. Zhang [86] proposed to
generalize the structure of Σ-norms which had appeared recently. The
authors defined the C-norm kC(G) of a finite group G as the intersection
of the normalizers of all subgroups of the group G which do not belong to
the class C
kC(G) =
⋂
H /∈C
NG(H)
provided that kC(G) = G, if G ∈ C. With this approach Baer norm N(G)
can be considered as the norm kC(G), where C is the class of groups
of order 1. Groups with kC(G) = G are called C-Dedekind. In [86] the
structure of non-nilpotent C-Dedekind groups for the class of nilpotent
groups is described. It is also shown that the groups, which C-norm is
not hypercentral, have a very restricted structure. The authors gave the
classification of nilpotent classes closed under subgroups, quotient groups
62 Generalized norms of groups
and direct products of groups of coprime orders, and showed that the
known classifications can be deduced from this one.
Proposition 3.1 ([86]). If kC(G) contains a non-central chief factor of
G, then kC(G) contains exactly one non-central chief factor (in any chief
series through kC(G) of a group G) and if p is a prime divisor of the
order of this chief factor, then Hall p′-subgroup of G is C-group and G
has nilpotency class at most 3.
Consider also R. Laue’s research [87]. He dealt with a subgroup close
to the Σ-norm
A(Σ) =
⋂
X∈Σ
NAut(G)(X),
which consists of automorphisms that normalize every Σ-subgroup of a
group G.
4. Generalized norms of different systems of Abelian and
non-cyclic subgroups
The narrowing of a system Σ of all subgroups of the group G to the
system of all Abelian and all cyclic subgroups does not lead to extension
of the concept of the norm N(G). However, when the system Σ is the
system of all non-cyclic subgroups (provided that such subgroups exist
in the group), then such Σ-norm (let’s call it the norm of non-cyclic
subgroups) differs from the norm N(G) in a general case. The opportunity
to study the norm of non-cyclic subgroups was provided by F. M. Lyman’s
research [88–90]. He received a description of some classes of non-Abelian
groups in which all non-cyclic subgroups are normal. These groups were
called H-groups (Hp-groups in the case of p-groups).
The concept of the non-cyclic norm NG of a group as the intersection
of the normalizers of all non-cyclic subgroups of the group was introduced
by F. M. Lyman in 1997 [91], where he studied infinite groups, in which
a non-cyclic norm is locally-graded and has a finite index.
Proposition 4.1 ([91]). In the group G a non-cyclic norm is locally-
graded and has a finite index if and only if the group G is central-by-finite.
In addition, it was proved that for the condition 1 < |G/NG| < ∞
in the class of infinite locally finite groups the non-cyclic norm NG is
Dedekind, and in the class of non-periodic locally soluble-by-finite groups
it is Abelian [91].
M. Drushlyak, T. Lukashova, F. Lyman 63
The study of the non-cyclic norm was continued by F. M. Lyman and
T. D. Lukashova in [92–96], where the authors characterized the structure
of wide classes of groups, which non-cyclic norm is non-Dedekind. Since
O. Yu. Olshansky infinite groups [97] exist, periodic groups were considered
by the authors provided their local finiteness. O. Yu. Olshansky infinite
groups are groups, all subgroups of which are cyclic and which are the
norms of their non-cyclic subgroups. Thus, in [98] it was proved that the
class of infinite locally finite p-groups (p 6= 2), in which a non-cyclic norm
NG is non-Abelian, coincides with the class of non-Abelian p-groups, all
non-cyclic subgroups of which are normal. At the same time, there are
infinite locally finite 2-groups which have a proper non-Dedekind norm of
non-cyclic subgroups. The structure of locally finite p-groups (p is prime),
which non-cyclic norm is non-Dedekind, is described in [92–94].
Proposition 4.2 ([92]). Locally finite p-groups (p 6= 2), which have
non-Abelian non-cyclic norm NG, are groups of the following types:
1) G is an Hp-group, NG = G;
2) G = (〈x〉 × 〈b〉)λ〈c〉, |x| = pn, n > 1, |b| = |c| = p, [b, c] = xpn−1
,
[x, c] = xpn−1αbβ, (β, p) = 1; NG = (〈xp〉 × 〈b〉)λ〈c〉;
3) G = 〈x〉〈b〉, |x| = pk, |b| = pm, m > 1, k > m + r, Z(G) = 〈xpr+1
〉 ×
〈bpr+1
〉, 1 6 r 6 m − 1, [x, b] = xpk−r−1sbpm−1t, (s, p) = 1, NG =
〈xpr
〉λ〈b〉.
Proposition 4.3 ([93,94]). Locally finite 2-groups G with a non-Dedekind
non-cyclic norm NG are groups of the following types:
1) G is a non-Hamiltonian H̄2-group, G = NG;
2) G = (A × 〈b〉)λ〈c〉λ〈d〉, A is a quasicyclic 2-group, [A, 〈c〉] = 1,
|b| = |c| = |d| = 2, d−1ad = a−1 for any element a ∈ A, [b, c] =
[d, b] = [d, c] = a1, a1 ∈ A, |a1| = 2; NG = (〈a〉 × 〈b〉)λ〈c〉, where
a ∈ A, |a| = 4;
3) G = (A × H)〈d〉 A is a quasicyclic 2-group, d2 = a1 ∈ A, | a1 |=
2, d−1ad = a−1 for any element a ∈ A, H = 〈h1, h2〉, |h1| =
|h2| = 4, h2
1 = h2
2 = [h1, h2], [d, h1] = a1, [d, h2] = 1; NG =
〈h2〉λ〈h1a〉, |a| = 4, a ∈ A, | a |= 4;
4) G = (〈x〉 × 〈b〉)λ〈c〉λ〈d〉, |x| = 2n,n > 2, |b| = |c| = |d| = 2, [x, c] =
1, d−1xd = x−1, [b, c] = [d, b] = [d, c] = x2n−1
; NG = (〈x2n−2
〉 ×
〈b〉)λ〈c〉;
5) G = (〈x〉λ〈b〉)λ〈c〉, |x| = 2n, n > 3, |b| = |c| = 2, [x, c] = x±2n−2
b,
[x, b] = x2n−1
; NG = (〈x2〉 × 〈b〉)λ〈c〉;
64 Generalized norms of groups
6) G = 〈x〉λH, |x| = 2n, n > 2, H = 〈h1, h2〉, |h1| = |h1| = 4, h2
1 =
h2
2 = [h1, h2], [〈x〉, H] = 〈x2n−1
〉; NG = 〈x2〉 × H;
7) G = (〈x〉 × H)〈y〉, |x| = 2n, n > 2, H = 〈h1, h2〉, |h1| = |h2| =
4, h2
1 = h2
2 = [h1, h2], y2 = x2n−1
, [y, h2] = 1, [y, h1] = y2, y−1xy =
x−1; NG = 〈h2〉λ〈h1x2n−2
〉;
8) G = 〈x〉〈b〉, |x| = 2k, |b| = 2m, m > 1; if m = 1, then k = 3,
[x, b] = x2 and NG = 〈x2〉λ〈b〉; if m > 1, then k > m + r, 1 6 r 6
m − 1, Z(G) = 〈x2r+1
〉 × 〈b2r+1
〉, [x, b] = x2k−r−1sb2m−1t, 0 < s <
2, 0 6 t < 2, (k > 3 and t = 0 if m = 2); NG = 〈x2r
〉λ〈b〉.
Non-primary locally finite groups with a non-Dedekind non-cyclic
norm were studied in [93,105]. It was found out that infinite locally finite
non-primary groups with such restrictions on the norm NG are locally
nilpotent.
Proposition 4.4 ([93]). Infinite locally finite non-primary groups with a
non-Dedekind non-cyclic norm NG, are locally nilpotent and are groups
of the following types:
1) G is an infinite non-primary non-Hamiltonian H̄–group, G = NG;
2) G = G2 × 〈y〉, G2 is a group of one type 2) or 3) of proposition 4.3,
(|y|, 2) = 1; NG = NG2
× 〈y〉.
Thus, a locally finite group, which non-cyclic norm NG is non-nilpotent,
is finite.
Developing the study of locally finite groups with a non-Dedekind
non-cyclic norm, in [95] it was proved that finite nilpotent groups with
such restrictions are groups of the type
G = Gp × 〈y〉,
where Gp is a Sylow p-subgroup of a group G and a finite group with a
non-trivial norm NGp
, (|y|, p) = 1. In addition, if the non-cyclic norm NG
is non-nilpotent in the class of locally finite groups, then all non-cyclic
subgroups in a group are normal.
Non-periodic locally soluble-by-finite groups with a non-Dedekind
non-cyclic norm are considered in [96].
Proposition 4.5 ([96]). Any non-periodic locally soluble-by-finite group
G that has a non-Dedekind non-cyclic norm NG is H-group and G = NG.
Note that locally finite or non-periodic locally soluble-by-finite groups
with a non-Dedekind norm of non-cyclic subgroups are soluble and their
degree of solvability does not exceed 3 according to the results of [92–96].
M. Drushlyak, T. Lukashova, F. Lyman 65
Zh. Shen, W. Shi, J. Zhang [98,99] studied the properties of the norm
NG of non-cyclic subgroups in the class of finite groups and its influence
on the group. The authors proved that the norm of non-cyclic subgroups
of a finite group is soluble. Note that this proposition is a direct corollary
from the description of finite H-groups (see [88–90]). It was also proved
that a finite group is soluble if all its elements of prime order are contained
in norm NG of non-cyclic subgroups. In addition, it was found out that
the derived subgroup is nilpotent if all elements of prime order or of order
4 of a group are contained in NG [98].
Proposition 4.6 ([98]). A finite group has a nilpotent derived subgroup
if and only if a derived subgroup of a quotient group through norm NG is
also nilpotent.
The study of infinite groups with given restrictions on normalizers of
different systems Σ of infinite subgroups have been the subject matter
of many theoretical-group researches for a long time. Therefore, when
considering infinite groups with restrictions on Σ-norm, it is naturally to
choose one of systems of infinite subgroups as a system Σ.
In this context, in the study of Σ-norms of infinite groups F. M. Lyman
and T. D. Lukashova [96,100–102] considered systems of all infinite, all
infinite Abelian and all infinite cyclic subgroups, provided that these
systems are non-empty. These Σ-norms were denoted as follows: NG(∞)
is the norm of infinite subgroups of a group G; NG(A∞) is the norm of
infinite Abelian subgroups of a group G; NG(C∞) is the norm of infinite
cyclic subgroups of a group G.
If the group G coincides with one of these Σ-norms, then all Σ-
subgroups are normal in it. Infinite non-Abelian groups with the property
NG(∞) = G and NG(A∞) = G (if such subgroups exist in them) were
studied by S. M. Chernikov [103, 104] and called INH-groups and IH-
groups respectively.
Restrictions, which these Σ-norms satisfied, were non-Dedekindness of
Σ-norm or finiteness of its index in the group. The following proposition
gives sufficient conditions of Dedekindness of each of these norms.
Proposition 4.7 ([100]). In non-periodic groups the norm NG(∞) of
infinite subgroups, the norm NG(A∞) of Abelian infinite subgroups, the
norm NG(C∞) of infinite cyclic subgroups are Dedekind in each of the
following cases:
1) G is a torsion free group or a mixed group without involution;
66 Generalized norms of groups
2) the center of a group G contains elements of infinite order;
3) G is central-by-finite;
4) these norms are finite;
5) G contains a subgroup M from the system Σ such that M
⋂
NG(Σ) =
E.
The problem of the relations between these norms in non-periodic
groups is quite interesting. The following relation is derived from the
above definitions
Z(G) ⊆ N(G) ⊆ NG(∞) ⊆ NG(A∞) ⊆ NG(C∞).
So the natural question is: under what conditions do these norms coin-
cide? The following proposition gives the answer (in terms of sufficient
conditions).
Proposition 4.8 ([100]). In a non-periodic group G the equality takes
place
N(G) = NG(∞) = NG(A∞) = NG(C∞)
provided that at least one of the statements takes place:
1) the center of a group G contains elements of infinite order;
2) G is a torsion free group;
3) G is central-by-finite.
Infinite groups with restrictions on the norm NG(∞) of infinite sub-
groups were studied in [100]. It turned out that non-periodic groups,
which norm NG(∞) has a finite index, are mixed and are finite extensions
of their centres.
It was also proved that the norm NG(∞) of infinite subgroups of the
non-periodic group is Abelian and coincides with the center of the group,
if it contains elements of infinite order. This result generalizes Baer’s
theorem [10] on the coincidence of the norm N(G) of the group and its
center in the case of a non-periodic norm N(G). Infinite locally finite
groups, which norm NG(∞) is non-Dedekind, are a finite extension of a
quasicyclic subgroup, which is a divisible part of the norm NG(∞) [101].
The structure of non-periodic groups, which norm NG(A∞) of infi-
nite Abelian subgroups is IH-group, are characterized by the following
proposition.
Proposition 4.9 ([96]). A non-periodic group G has non-Abelian norm
NG(A∞) of infinite Abelian subgroups, if and only if all elements of infinite
M. Drushlyak, T. Lukashova, F. Lyman 67
order of the group G generate Abelian normal subgroup D that contains
every infinite Abelian subgroup of a group G and there is an element b
of order 2 or 4, such that b−1db = d−1 for an arbitrary element d ∈ D.
Moreover NG(A∞) = D〈b〉.
A natural generalization of Baer norm for non-periodic groups is the
norm NG(C∞) of infinite cyclic subgroups. The study of this norm and
its influence on properties of the group was started by F. M. Lyman and
T. D. Lukashova in [102]. It was proved that the norm NG(C∞) coincides
with the center of the group in torsion free groups, and any finite over the
norm NG(C∞) torsion free group is Abelian. The following proposition
characterizes the properties of the group that has non-Abelian norm
NG(C∞).
Proposition 4.10 ([102]). A non-periodic group G has non-Abelian
norm NG(C∞), if and only if all elements of infinite order of the group G
generate an Abelian normal subgroup A and there is an element b of order
2 or 4, such that b−1ab = a−1 for an arbitrary element a ∈ A. Moreover
NG(C∞) = A〈b〉.
Let’s note, if the norm NG(A∞) is non-Abelian in a non-periodic
group, then the norm NG(C∞) of infinite cyclic subgroups is non-Abelian.
Moreover, in this case NG(C∞) = NG(A∞). The following example shows
that non-periodic groups, which norm NG(C∞) is non-Abelian and norm
NG(A∞) is Abelian, exist.
Example 4.1. G = (〈a〉λ〈b〉) × C, |a| = ∞, |b| = 2, C is an infinite
elementary Abelian 2-group, b−1ab = a−1.
It is easy to prove that NG(A∞) = C is Abelian, NG(C∞) = G and
NG(A∞) 6= NG(C∞).
The following proposition characterizes the conditions when the norm
NG(A∞) coinsides with the norm NG(C∞) in a non-periodic group G
(provided that the subgroup NG(C∞) is non-Abelian).
Proposition 4.11. Let G be a non-periodic group, which norm NG(C∞)
of infinite cyclic subgroups is non-Abelian. Subgroups NG(C∞) and
NG(A∞) coincide, if and only if NG(C∞) is central-by-finite and contains
every infinite Abelian subgroup of G.
In connection with the existence of O. Yu. Olshansky groups, periodic
groups with non-Dedekind norm of infinite Abelian subgroups were studied
68 Generalized norms of groups
under the condition of their local finiteness. In [100] it was proved that
such groups satisfy the minimal condition for subgroups, if and only if
subgroup NG(A∞) satisfies this condition. Moreover, if NG(A∞) is a
group with minimal condition for subgroups, then G is a finite extension
of its divisible part and therefore [G : NG(A∞)] < ∞.
Note that the norm N(G) can be considered as the intersection of the
normalizers of all cyclic subgroups. In this connection it is naturally to
consider Σ-norm, where Σ consists of all cyclic subgroups of non-prime
order of this group. Such a norm was studied by T. D. Lukashova and
M. G. Drushlyak [105] in the class of non-periodic groups and was called
the norm NG(Cp̄) of cyclic subgroups of non-prime order of the group G.
It is clear that all cyclic subgroups of compound or infinite order are
normal in non-periodic group G, which coincides with the norm NG(Cp̄).
Such non-Dedekind groups were studied by T. G. Lelechenko, F. M. Lyman
[106] and were called almost Dedekind groups.
Since the norm NG(Cp̄) normalizes each infinite cyclic subgroup of a
group G, NG(Cp̄) ⊆ NG(C∞) in non-periodic groups. It turns out that
these norms coincide, if the norm NG(Cp̄) of cyclic subgroups of non-prime
order is non-Abelian.
Proposition 4.12 ([105]). Any non-periodic group G that has non-
Abelian norm NG(Cp̄) of cyclic subgroups of non-prime order is almost
Dedekind and coincides with this norm. Moreover G = Aλ〈b〉, where A is
a normal Abelian subgroup which contains all elements of prime order of
group G, |b| = 2, b−1ab = a−1 for an arbitrary element a ∈ A.
In 2004 in [107,108] F. M. Lyman and T. D. Lukashova introduced one
more Σ-norm, where Σ is a system of all Abelian non-cyclic subgroups of
a group. This Σ-norm was called the norm of Abelian non-cyclic subgroups
of a group G and denoted by NA
G . It is clear if the group G coincides
with the norm NA
G then all Abelian non-cyclic subgroups are normal in it
(assuming the existence of at least one of such a subgroup). Non-Abelian
groups with this property were fully described in [107,109,110] and called
HA-groups (HAp-groups in the case of p-groups).
In [92,107,114] infinite locally finite p−groups (p is an arbitrary prime),
which norm NA
G is non-Dedekind, are considered. The authors obtained
a complete description of such groups and proved that if the norm NA
G
is infinite and non-Dedekind, then all Abelian non-cyclic subgroups are
normal in a group, that is in this case G = NA
G . It was also proved that
M. Drushlyak, T. Lukashova, F. Lyman 69
locally finite p-groups with non-Dedekind norm NA
G are finite extensions
of a quazicyclic group. In particular, the following propositions take place.
Proposition 4.13 ([107]). Infinite 2-groups with non-Dedekind norm
NA
G of Abelian non-cyclic subgroups are groups of one of the following
types:
1) G is an infinite HA2-group, NA
G = G;
2) G = (A × 〈b〉)λ〈c〉λ〈d〉, where A is a quasicyclic 2-group,|b| =
|c| = |d| = 2, [A, 〈c〉] = 1, [b, c] = [b, d] = [c, d] = a1 ∈ A, |a1| =
2, d−1ad = a−1 for any element a ∈ A; NA
G = NG = (〈a2〉 ×
〈b〉)λ〈c〉, a2 ∈ A, |a2| = 4;
3) G = A〈y〉H, where A is a quasicyclic 2-group, [A, H] = E, H =
〈h1, h2〉, |h1| = 4, h2
1 = h2
2 = [h1, h2], |y| = 4, y2 = a1 ∈ A, y−1ay =
a−1 for any element a ∈ A, [〈y〉, H] ⊆ 〈a1〉 × 〈h2〉; NA
G = 〈a2〉 ×
H, a2 ∈ A, |a2| = 4, NG = 〈h2〉λ〈h1a2〉.
Proposition 4.14 ([108]). Infinite locally finite p-groups (p 6= 2), which
norm NA
G of Abelian non-cyclic subgroups is non-Dedekind, are HAp-
groups and G = NA
G = NG.
Let’s note that Proposition 4.14 fails in the case of infinite locally
finite 2-groups: there are infinite 2-groups with finite non-Dedekind norm
NA
G of Abelian non-cyclic subgroups, which may not coincide with the
norm NG.
The study of finite p-groups for certain restrictions on the norm NA
G
of Abelian non-cyclic subgroups were continued by M. G. Drushlyak,
T. D. Lukashova and F. M. Lyman in [112,113]. In particular, in [112] the
structure of finite p-groups (p 6= 2) with a non-Abelian norm of Abelian
non-cyclic subgroups was completely described, in [113] the structure of
finite 2-groups with a non-cyclic centre and non-Dedekind norm NA
G of
Abelian non-cyclic subgroups was described. It is also proved that an
arbitrary 2-group with a non-cyclic centre and a non-Dedekind norm NA
G
does not contain a quaternion subgroup, if and only if the norm NA
G does
not contain such a subgroup. In this case the norm NA
G coinsides with
the norm NG [113].
The following proposition clarifies the result of [92] on the coincidence
of norms NA
G and NG for infinite locally finite p-groups (p 6= 2) under the
condition that the subgroup NG is non-Abelian.
Proposition 4.15 ([112]). If either norm NA
G or NG is non-Abelian,
then NG = NA
G in the class of locally finite p-groups (p 6= 2).
70 Generalized norms of groups
The Proposition 4.15 leads to the conclusion that any finite p-group
(p 6= 2) with non-Abelian norm NA
G is a group of one of the types of 1)-3)
of Proposition 4.2.
Proposition 4.16 ([113]). Finite 2-groups with a non-cyclic centre and
a non-Dedekind norm NA
G of Abelian non-cyclic subgroups is a group of
the following types:
1) G is a non-Dedekind non-metacyclic HA2–group with a non-cyclic
center, G = NA
G ;
2) G = H · Q is a product of a quaternion group of order 8 and
a generalized quaternion group; H = 〈h1, h2〉, |h1| = |h2| = 4,
[h1, h2] = h2
1 = h2
2, Q = 〈y, x〉, |y| = 2n, n > 3, |x| = 4, y2n−1
=
x2, x−1yx = y−1, [Q, H] ⊆ 〈x2, h2
1〉; NA
G = 〈y2n−2
〉 × H;
3) G = 〈x〉〈b〉, |x| = 2k, |b| = 2m, m > 2, k > m + r, 1 6 r < m −
1, Z(G) = 〈x2r+1
〉 × 〈b2r+1
〉, [x, b] = x2k−r−1sb2m−1t, (s, 2) = 1, 0 6
t < 2; NA
G = NG = 〈x2m−1
〉λ〈b〉.
Developing the study of finite 2-groups T. D. Lukashova, F. M. Lyman
and M. G. Drushlyak obtained a structural description of groups with a
cyclic center and a non-metacyclic non-Dedekind norm NA
G .
Proposition 4.17. Finite 2-groups with a non-metacyclic non-Dedekind
norm NA
G of Abelian non-cyclic subgroups and a cyclic centre are groups
of the following types:
1) G is a non-metacyclic non-Hamiltonian HA2-group with a cyclic
center, G = NA
G ;
2) G = (〈x〉λ〈c〉)λ〈b〉, |x| = 2n, n > 3, |b| = |c| = 2, [x, b] = x±2n−2
c,
[b, c] = [x, c] = x2n−1
, NA
G = NG = (〈x2〉 × 〈c〉)λ〈b〉;
3) G = (〈x〉 × 〈b〉)λ〈c〉λ〈d〉, |x| = 2n, n > 2, |b| = |c| = |d| = 2,
[x, c] = [x, b] = 1, [b, c] = [c, d] = [b, d] = x2n−1
, d−1xd = x−1,
NA
G = NG = (〈x2n−2
〉 × 〈b〉)λ〈c〉;
4) G = (〈c〉λH)〈y〉, H = 〈h1, h2〉, |h1| = |h2| = 4, h2
1 = h2
2 = [h1, h2],
|c| = 4, [c, h2] = 1, [c, h1] = c2, y2 = h1, [y, h2] = c2h2
1, [y, c] =
h±1
2 ; NA
G = 〈c〉λH.
The question of the structure of finite 2-groups with a cyclic center,
in which the norm NA
G is a metacyclic non-Dedekind group, is still open.
Study of the influence of the properties of the norm of Abelian non-
cyclic subgroups on the properties of the group was continued in [114],
where infinite periodic groups, which norm NA
G is non-Dedekind and
M. Drushlyak, T. Lukashova, F. Lyman 71
locally nilpotent, were considered. It was proved that such groups satisfy
the minimal condition for Abelian subgroups and are Chernikov groups.
Proposition 4.18 ([114]). An infinite periodic locally nilpotent group G
has a non-Dedekind norm of Abelian non-cyclic subgroups, if and only if
G = Gp × Gp′ ,
where Gp is an infinite Sylow p-subgroup of a group G with a non-Dedekind
norm NA
Gp
of Abelian non-cyclic subgroups (where p ∈ π(G)) and Gp′ is
a finite cyclic or finite Hamiltonian p′-subgroup, all Abelian subgroups of
which are cyclic, and NA
G = NA
Gp
× Gp′.
If G is a locally finite, not locally nilpotent group which has an infinite
locally nilpotent non-Dedekind norm NA
G , then G = Gp ⋋ H, where Gp
is an infinite HAp-group, which coincides with a Sylow p-subgroup of a
norm NA
G , and H is a finite group, all Abelian subgroups of which are
cyclic, (|H|, p) = 1. In addition, the structure of infinite locally finite
non-nilpotent groups, which norm NA
G is finite non-Dedekind nilpotent
subgroup, was described.
Study of the norm NA
G of Abelian non-cyclic subgroups in the class of
non-periodic groups were continued by M. G Drushlyak and F. M. Lyman.
In [115, 116] non-periodic groups with non-Dedekind norm of Abelian
non-cyclic subgroups depending on the presence [115] or the absence [116]
of a free Abelian subgroup of rank 2 were considered.
Proposition 4.19 ([115]). If a non-periodic group G contains a free
Abelian subgroup of rank 2, its norm NA
G of Abelian non-cyclic subgroups
is non-Dedekind, contains an Abelian non-cyclic subgroup and a finite
Abelian, normal in G, subgroup F and the centralizer CG(F ) contains all
elements of infinite order of a group, then NA
G = NG(C∞) = B〈d〉, where
B is the Abelian subgroup generated by all elements of infinite order of
the group G, |d| = 2 or |d| = 4, d2 ∈ B, d2 is a unique involution in G
and d−1bd = b−1 for an arbitrary element b ∈ B.
It was also proved that the non-periodic group G does not contain
free Abelian subgroups of rank 2, if its norm NA
G is non-Hamiltonian
HA-group and does not contain such subgroups.
In 2015 F. M. Lyman and T. D. Lukashova [117] considered one more
generalization of the concept of the norm of the group – the norm Nd
G
of decomposable subgroups, which is defined as the intersection of the
72 Generalized norms of groups
normalizers of all decomposable subgroups of the group. In the case when
the group does not contain any decomposable subgroups, we can assume
that Nd
G = G. The structure of locally soluble groups, in which a system
of decomposable subgroups is empty, as well as groups, in which each
decomposable subgroup is normal (groups with the condition Nd
G = G),
was described in [118].
It is clear that the group contains decomposable subgroups, if and
only if it contains decomposable Abelian subgroups. Therefore, the study
of the norm Nd
G of decomposable subgroups was conducted, regarding on
the existence of systems of decomposable Abelian subgroups in the group.
Thus the norm Nd
G of decomposable subgroups is closely related to the
norm NA
G of Abelian non-cyclic subgroups. In particular, in [117] it was
proved that these norms coinside in the class of locally finite p-groups.
The inclusion NA
G ⊇ Nd
G takes place and the case NA
G 6= Nd
G is achieved
in classes of finite non-primary groups, as well as in classes of infinite
periodic locally nilpotent non-primary groups.
Proposition 4.20 ([117]). A periodic locally nilpotent group G which
contains an Abelian non-cyclic subgroup has a non-Dedekind norm Nd
G of
decomposable subgroups, if and only if G is a locally finite p-group with a
non-Dedekind norm NA
G of Abelian non-cyclic subgroups.
Study of the influence of the norm Nd
G of decomposable subgroups on
the properties of the group were extended by the authors in the class of
non-periodic groups. In particular, in [119] the following was established.
Proposition 4.21 ([119]). Let G be a non-periodic group that has a
non-Dedekind norm Nd
G of decomposable subgroups. Then the following
propositions take place:
1) G does not contain decomposable subgroups if and only if the norm
Nd
G of the group does not contain them;
2) G contains a free Abelian subgroup of rank r > 2, if and only if the
norm Nd
G contains a free Abelian subgroup of such a rank;
3) G contains a non-primary Abelian subgroup, if and only if the norm
Nd
G of the group contains subgroups with this property;
4) any decomposable Abelian subgroup of a group G is mixed, if and
only if any decomposable Abelian subgroup of its norm Nd
G is mixed.
It was also proved that in the class of non-periodic locally soluble
groups only one of the inclusions NA
G ⊇ Nd
G or NA
G ⊆ Nd
G takes place,
provided that at least one of these norms is non-Dedekind and the norm
M. Drushlyak, T. Lukashova, F. Lyman 73
Nd
G is infinite. The following examples confirm that the condition of the
infiniteness of the norm Nd
G is essential.
Example 4.2 ([119]). G = (〈a〉⋋B)⋋〈c〉, where |a| = p, p is a prime (p 6=
2), B is a group isomorphic to an additive group of q-adic numbers, q /∈
{2, p} , B = B1〈x〉, x2 ∈ B1, x−1ax = a−1, [B1, 〈a〉] = E, |c| = 2, [c, a] =
1, c−1bc = b−1 for any element b ∈ B.
In this group all periodic decomposable subgroups are of order 2p
and are groups of the type 〈amcbk
1〉, where b1∈ B1, k ∈ {0, 1} , (m, p) = 1.
Accordingly, all non-periodic decomposable subgroups are mixed and
contained in the group B1 × 〈a〉 and therefore are normal in G. Since
NG(〈amcbk
1〉) = 〈amcbk
1〉, Nd
G = 〈a〉.
On the other hand, G does not contain periodic Abelian non-cyclic
subgroups and all mixed Abelian subgroups contain 〈a〉 and are subgroups
of the group (B1×〈a〉) and therefore are normal in G. Moreover, all Abelian
non-cyclic subgroups of rank 1 are contained either in the subgroup B, or
in subgroups conjugated with it g−1Bg, g ∈ G, or in the group (B1 × 〈a〉).
Let’s consider an infinite sequence of subgroups in B1:
〈b1〉 ⊂ 〈b2〉 ⊂ · · · ⊂ 〈bn〉 ⊂ · · · ,
|b1| = ∞, b
αn+1
n+1 = bn, αn+1 ∈ N, (αn+1, p) = 1 for n = 1, 2, . . . Since
the isolator A of the subgroup 〈ab1〉 is non-cyclic (because the root of
an arbitrary degree coprime with p can be taken from the element a),
NG(A) = 〈a, B1〉. NA
G = B1 and Nd
G ∩ NA
G = E by NG(B) = B ⋋ 〈c〉.
Example 4.3 ([119]). G = (〈a〉⋋B)⋋〈c〉, where |a| = p, p is a prime (p 6=
2), B is a group isomorphic to an additive group of p-adic numbers, B =
B1〈x〉, x2 ∈ B1, x−1ax = a−1, [B1, 〈a〉] = E, |c| = 2, [c, a] = 1, c−1bc =
b−1 for any element b ∈ B.
As in Example 4.2 in this group the norm of decomposable subgroups
is Nd
G = 〈a〉. However, the norm of Abelian non-cyclic subgroups is
NA
G = (B1 ⋋ 〈c〉). This follows from the fact that for any non-identity
element y1 ∈ B1 the isolator of a subgroup 〈ay1〉 is cyclic, and therefore the
element c normalizes each Abelian non-cyclic subgroup of a group G. In
this case, the norm NA
G of Abelian non-cyclic subgroups is non-Dedekind
and Nd
G ∩ NA
G = E.
In 2005 F. Mari, F. de Giovanni [78] considered the concept of the
non-Abelian norm N∗(G) that is the intersection of normalizers of all
74 Generalized norms of groups
non-Abelian subgroups of the group. If N∗(G) = G, then all non-Abelian
subgroups are normal in the group. These groups were studied by G. M. Ro-
malis and N. F. Sesekin [120–122] and were called metahamiltonian.
Further metahamiltonian groups were studied by V. T. Nagrebezkiy
[123], O. A. Makhnev [124], S. M. Chernikov [125], M. M. Semko and
M. F. Kuzennyi [126].
In [78] the results that generalize Schur theorem [127] on finiteness of
derived subgroups in central-by-finite groups were offered.
Proposition 4.22 ([78]). If G is a locally graded group and the quotient
group G/N∗(G) is finite, then a derived subgroup G′ is finite.
Conclusion
The authors make a conclusion that the study of different Σ-norms
and properties of groups with respect on properties of their Σ-norms is a
very important field in the group theory. Nowadays the research of groups
that differ from their Σ-norms as well as groups that have a non-Dedekind
Σ-norm becomes possible, because the structure of groups that coincide
with Σ-norms is well known in many cases. Therefore it will give the
opportunity to extend the known classes of generalized Dedekind groups
and will allow to study groups with restrictions on the normalizers of
different systems of subgroups more effectively.
There are still a number of problems in the study of groups with
generalized norms:
– the study of groups that coincide with their Σ-norms;
– the study of groups that have identity Σ-norms or their Σ-norms
coinside with the center;
– the study of groups that have non-central Dedekind Σ-norms;
– the study of groups that have non-Dedekind Σ-norms;
– the study of infinite groups that have Σ-norms of finite index.
The solution of these problems will significantly expand the base of
the modern group theory.
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Contact information
M. Drushlyak,
T. Lukashova,
F. Lyman
Sumy State University, Romenska Str. 87, 40002
Sumy, Ukraine
E-Mail(s): marydru@mail.ru,
tanya.lukashova2015@gmail.com,
mathematicsspu@mail.ru
Received by the editors: 22.01.2016
and in final form 24.02.2016.
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