Algebra and Discrete Mathematics SURVEY ARTICLE
Volume 15 (2013). Number 2. pp. 161 – 173
c© Journal “Algebra and Discrete Mathematics”
On Baer-Shemetkov’s decomposition in modules
and related topics
V. V. Kirichenko, L. A. Kurdachenko,
A. A. Pypka, I. Ya. Subbotin
Dedicated to the memory of an outstanding mathematician
Leonid A. Shemetkov
Abstract. This article is dedicated to the memory of an
outstanding algebraist Leonid A. Shemetkov. His ideas and results
not only shaped modern soluble finite group theory, but significantly
influenced other branches of algebra. In this article, we traced the
influence of L.A. Shemetkov’s ideas on some areas of modules theory
and infinite groups theory.
On March 24, 2013, one of the greatest experts in the theory of
finite groups Leonid A. Shemetkov passed away. He was one of those
people who developed and shaped the modern theory of finite non-simple
groups. His contribution to this field of algebra is hard to overestimate.
L.A. Shemetkov productively and intensively investigated finite groups,
and his ideas and influence on its development have been reflected in
numerous works of his many students and followers. Led by him, the
Gomel School is one of the world leading scientific schools in group
theory. His outstanding results have been reflected in books and review
articles. L.A. Shemetkov’s research affected significantly finite group
theory development, and his ideas have been expanded to some important
areas of infinite groups as well. Not only had the people directly related to
his school felt L.A. Shemetkov’s influence. L.A. Shemetkov and his school
had tight relations with Ukrainian algebraists. Along with his results,
2010 MSC: 13E10, 20F17, 20F19.
Key words and phrases: formation, modules over group rings, artinian modules,
direct decomposition, complement, supplement.
162 On Baer-Shemetkov’s decomposition in modules
direct communications with him and his views on the problems have helped
to shape some productive approaches to infinite groups study. In the
current paper, we set out to do a review of the well known Shemetkov’s
results; it has been done already. We want to reflect something else.
Here we would like to consider (and to show in the process of their
development) some cross-related areas of group theory and module over
group rings theory, the shaping of which L.A. Shemetkov had a great
influence. The theory of infinite groups is significantly distinct from the
theory of finite groups. The expanding of the results obtained for finite
groups to infinite groups often possible only for sufficiently narrow classes
of infinite groups, or instead of a single counterpart it leads to a series
of results. And formation theory plays an important role here. In finite
groups, this theory has been is rapidly developing for the last 40 years.
L.A. Shemetkov was one of those who laid the foundation of formation
theory (see the book [17]). The intensive development of finite group
formation theory caused a natural interest among professionals working
in infinite groups. A whole series of works devoted to the extension of
the main results of formations of finite groups to locally finite groups
have been published. Thus the concept of the formation was substantially
narrowed, namely the condition (F2) (please see below) was replaced by
the following:
if {Hλ|λ ∈ Λ} is a family of normal subgroups, then G/
⋂
λ∈Λ Hλ ∈ X.
There was quite a long time when the theme the purpose of which was
almost a literal transfer of the main results of the theory of formations
obtained for finite groups to certain classes of infinite groups has evolved.
The transfer of the results of formation theory was possible only where
there exists a very extensive Sylow theory (for example, in periodic almost
locally nilpotent groups, periodic linear groups, and periodic FC-groups).
Under the mentioned narrowed definition, many natural classes of groups,
such as, finite groups, (locally) soluble groups, (locally) nilpotent groups,
π-groups, etc., are not the formations. We will use here the classical
definition of a formation, i.e. namely the definition of a formation that
has arisen and works in finite groups.
A class of groups X is said to be a formation of groups if the following
conditions hold:
(F1) If G ∈ X and H is a normal subgroup of G, then G/H ∈ X;
(F2) If H,K are the normal subgroups of G such that G/H ∈ X,
G/K ∈ X, then G/(H ∩K) ∈ X.
Kirichenko, Kurdachenko, Pypka, Subbotin 163
For these formations, some of the commonly used in finite groups ap-
proaches work very well in infinite groups. They began to be implemented
first for the particular formation, i.e. for the formation F of all finite
groups, and then for other formations which play an important role in the
theory of infinite groups. Here we want to show some areas where it is
effectively implemented. One of this areas, which are so remote from the
area of finite groups, is artinian modules over group rings. An important
aspect of the theory of Artinian modules is a problem of finding their
natural direct decompositions. One of the first results in this area is the
following well-known Fitting’s Lemma. Here we present it in the following
form.
Theorem 1 (Fitting). Let R be a ring, G a finite nilpotent group and
A an RG-module. If A has a finite composition length, then A = Z ⊕ E,
where every RG-chief factor U/V of Z (respectively of E) satisfies the
condition G = CG(U/V ) (respectively, G 6= CG(U/V )).
Now we need some basic concepts. One of the first basic concepts
of the formation theory is the concept of an X-central factor. We will
formulate them in parallel for groups and modules.
Let G be a group, U, V be normal subgroups of G, such that
U ≤ V . The factor V/U is called X-central (respectively, X-eccentric),
if G/CG(V/U) ∈ X (respectively, G/CG(V/U) 6∈ X). Clearly, if X = I
is a class of all identity groups, then we come to the concept of central
factor. Let G be a finite group and suppose that G has a series of normal
subgroups 〈1〉 = H0 ≤ H1 ≤ ... ≤ Hn−1 ≤ Hn = G whose factors are
X-central. If g ∈ H1, then gG = {gx|x ∈ G} ⊆ H1. If X is a formation,
then G/CG(gG) ∈ X and we come to the following important concept.
Let G be a group and X be a class of groups. Put
XC(G) = {x ∈ G|G/CG(gG) ∈ X}.
If X is a formation, then it is not hard to prove that XC(G) is a character-
istic subgroup of G. A subgroup XC(G) is said to be the XC-center of a
group G. If X = I is a class of all identity groups, then XC(G) = ζ(G) is
an ordinary center of G. A group G is called an XC-group if G = XC(G).
Thus we see that the class of XC-groups is a natural extension of the
class of abelian groups. If X = F then XC-group is called an FC-group.
Starting from the XC-center of a group G, we construct the upper
XC-central series of G as
〈1〉 = B0 ≤ B1 ≤ ... ≤ Bα ≤ Bα+1 ≤ ... ≤ Bγ ,
164 On Baer-Shemetkov’s decomposition in modules
where B1 = XC(A), Bα+1/Bα = XC(A/Bα) for all ordinals α < γ and
〈0〉 = XC(A/Bγ). The last term Bγ of this series is called the upper XC-
hypercenter of A and is denoted by XC∞(A). If A = XC∞(A), then A is
said to be XC-hypercentral; if γ is finite, then A is called XC-nilpotent.
Let X be a class of groups, G a group and
H(X) = {H|H is a normal subgroup of G such that G/H ∈ X},
GX = ⋂H(X) = ⋂H∈H(X) H.
A subgroupGX is called the X-residual of a group G. If X is a formation
and a family H(X) is finite (in particular, if G is finite), then G/GX ∈ X.
There are some analogs of these concepts for modules over group rings.
Let R be a ring, G a group and A an RG-module. If B,C are
RG-submodules of A such that B ≤ C, the factor C/B is said to be
X-central (respectively, X-eccentric) if G/CG(C/B) ∈ X (respectively,
G/CG(C/B) 6∈ X).
To rule out these factors, we define
XCRG(A) = {a ∈ A|G/CG(aRG) ∈ X}.
Observed at once, that if X is a formation of groups, then XCRG(A) is
an RG-submodule of A. The submodule XCRG(A) is called the XC−RG-
center of A (shorter, the XC-center of A). Started from XC-center, we
construct the upper XC −RG-central series of the module A as
〈0〉 = A0 ≤ A1 ≤ ... ≤ Aα ≤ Aα+1 ≤ ... ≤ Aγ ,
where A1 = XCRG(A), Aα+1/Aα = XCRG(A/Aα) for all ordinals α < γ
and XCRG(A/Aγ) = 〈0〉. The last term Aγ of this series is called the
upper XC − RG-hypercenter of A (in short, the XC-hypercenter of A)
and is denoted by XC∞RG(A). If A = XC∞RG(A), then A is said to be
XC −RG-hypercentral; if γ is finite, then A is called XC −RG-nilpotent.
We note that, if X = I is the class of all identity groups, then
XCRG(A) = ζRG(A) is called the RG-center of A and XC∞RG(A) = ζ∞RG(A)
is called the upper RG-hypercenter of A. If X = F is the class of all finite
groups, then XCRG(A) = FCRG(A) is called the FC-center of A and
XC∞RG(A) = FC∞RG(A) is called the upper FC-hypercenter of A.
An RG-submodule C of A is said to be X−RG-hypereccentric if it
has an ascending series
〈0〉 = C0 ≤ C1 ≤ ... ≤ Cα ≤ Cα+1 ≤ ... ≤ Cγ = C
Kirichenko, Kurdachenko, Pypka, Subbotin 165
of RG-submodules of A such that each factor Cα+1/Cα is an X-eccentric
simple RG-module for every ordinal α < γ.
If A is an artinian RG-module, then it has an ascending series of RG-
submodules whose factors are simple RG-modules. If X is an arbitrary
formation and G ∈ X, then the factors of this series can be located in
some different ways. L.A. Shemetkov indicated the formations (in the
finite groups case) where it will be the most convenient location.
Theorem 2 (L.A. Shemetkov [16]). Let G be a finite group and X be a
local formation. If p be a prime such that a Sylow p-subgroup P of GX is
abelian, then every chief p-factor of GX is X-eccentric. In particular, if
Sylow q-subgroups of GX are abelian for every prime q, then every chief
factor of GX is X-eccentric.
Suppose now that G includes a normal abelian subgroup A such that
G/A ∈ X (that is GX ≤ A). From some other results of this Shemetkov’s
paper (we will consider it a little bit later), one can derive the existence
of a subgroup C such that G = GXC and C ∩GX = 〈1〉. Put E = C ∩A,
then clearly E is normal in G and A = GX × E. By Theorem 2, every
chief factor of GX is X-eccentric. The same result was obtained in the
paper of R. Baer [1]. Observe, that if X is a local formation, then every
finite XC-nilpotent group belongs to X (see K. Doerk and T.O. Hawkes
[[2], Theorem IV.3.2]). Thus we come to the following concept.
We say that the RG-module A has the Baer-Shemetkov’s decom-
position for the formation X, or A has the Baer-Shemetkov’s X − RG-
decomposition if
A = XC∞RG(A)⊕ XE∞RG(A),
where XE∞RG(A) is the maximal X−RG-hypereccentric RG-submodule
of A. It is possible to prove that in this case, XE∞RG(A) includes every
X − RG-hypereccentric RG-submodule and, in particular, XE∞RG(A) is
defined uniquely.
If X = I, we will say about the Z-decomposition, whereas if X = F, we
will say about the F-decomposition. The first natural question for infinite
modules was the question of the conditions under which there is the
Z-decomposition, and the first work where this issue has been addressed
in the paper of B. Hartley and M.J. Tomkinson [7].
Theorem 3 (B. Hartley and M.J. Tomkinson [7]). Let G be a locally
nilpotent group and let A be a ZG-module. If A is Z-periodic and the p-
component of A are artinian as Z-modules (that is are Chernikov groups)
for every prime p, then A has the Z-decomposition.
166 On Baer-Shemetkov’s decomposition in modules
In this case, the factor-group G/CG(A) is actually hypercentral. The
following case, namely the case of artinian ZG-module over hypercentral
group G, has been considered by D.I. Zaitsev.
Theorem 4 (D.I. Zaitsev [18]). Let G be a hypercentral group and let
A be a ZG-module. If A is an artinian ZG-module, then A has the Z-
decomposition.
Note that this result can be easily extended to an artinian DG-module,
where D is a Dedekind domain and G is a hypercentral group.
The following natural step was the consideration of existence of the
F-decomposition for artinian modules. We notice first that in this de-
liberation the considered group has to be FC-hypercentral instead of
hypercentral. The first results in this direction have been obtained by
Zaitsev.
Theorem 5 (D.I. Zaitsev [19]). Let G be a hyperfinite locally soluble
group and A a ZG-module. If A is an artinian ZG-module, then A has
the F-decomposition.
Theorem 6 (D.I. Zaitsev [20]). Let G be an FC-hypercentral group and
let A be a ZG-module. If A has finite composition ZG-series, then A has
the F-decomposition.
The next result belongs to Z.Y. Duan, who considered the following
special case of FC-hypercentral groups.
Theorem 7 (Z.Y. Duan [3]). Let G be a locally soluble group having an
ascending series of normal subgroups, every factor of which is finite or
cyclic and let A be a ZG-module. If A is an artinian ZG-module, then A
has the F-decomposition.
The most general solution for the formation F was obtained by
L.A. Kurdachenko, B.V. Petrenko and I.Ya. Subbotin.
Theorem 8 (L.A. Kurdachenko, B.V. Petrenko, I.Ya. Subbotin [8]).
Let G be a locally soluble FC-hypercentral group, D a Dedekind domain
and A a DG-module. If A is an artinian DG-module, then A has the
F-decomposition.
These partial cases have played an important role in the study of the
general case of existence of Baer-Shemetkov’s X-decomposition. Actually,
Kirichenko, Kurdachenko, Pypka, Subbotin 167
its solution has allowed us to obtain solutions for many important forma-
tions X. To deal with this, it is convenient to split the entire study into
two cases: X includes F and X is a proper formation of finite groups.
A formation X is said to be overfinite if it satisfies the following
conditions:
(i) if G ∈ X and H is a normal subgroup of G having finite index, then
H ∈ X;
(ii) if G is a group, H is a normal subgroup of G such that G/H is
finite and H ∈ X, then G ∈ X;
(iii) I ≤ X.
Clearly, every overfinite formation always includes F. The most im-
portant examples of these formations are polycyclic groups, Chernikov
groups, soluble minimax groups, soluble groups of finite special rank, and
soluble groups of finite section rank. The most general result here is the
following
Theorem 9. Let G be a locally soluble FC-hypercentral group, D a
Dedekind domain and A be an artinian DG-module. If X is an overfinite
formation of groups, then A has the Baer-Shemetkov’s X-decomposition.
For the case D = Z, this result was obtained by L.A. Kurdachenko,
B.V. Petrenko and I.Ya. Subbotin in the paper [9]. In a general form it
was placed in the book [[11], Chapter 10]. It is worth mentioning that
since every overfinite formation X includes the formation of all finite
groups, every FC-hypercentral group is likewise XC-hypercentral.
Corollary 1. Let G be a locally soluble FC-hypercentral group, D a
Dedekind domain and A be an artinian DG-module. Then A has the
Baer-Shemetkov’s X-decomposition for the following formations X:
(i) X is the formation of all polycyclic groups;
(ii) X is the formation of all Chernikov groups;
(iii) X is the formation of all soluble minimax groups;
(iv) X is the formation of all soluble groups of finite special rank;
(v) X is the formation of all soluble groups of finite section rank.
In connection with Theorem 9, the following question naturally arises:
for what formations of finite groups does the Baer-Shemetkov’s decompo-
sition exist? Before dealing with it, we first notice that infinite groups
168 On Baer-Shemetkov’s decomposition in modules
behave badly with respect to some properties automatically satisfied by
finite groups. For example, suppose that X is a formation of finite groups
and G is a residually finite group. If G is finite, then G ∈ X. However, if G
is infinite, the situation can be totally different. To avoid this complication,
we consider the following formation of finite groups.
A formation X of finite groups is said to be infinitely hereditary
concerning a class of groups H if it satisfies the following condition:
(IH) whenever an H-group G is residually X-group, then every finite
factor-group of G belongs to X.
It is worth mentioning that many formations of finite groups are
infinitely hereditary concerning the class of FC-hypercentral groups. For
example we can mention finite abelian groups, finite nilpotent groups
of class at most c, finite soluble groups of derived length at most d,
finite soluble groups, and finite groups of exponent dividing n. Moreover,
all groups from these five examples, finite nilpotent groups, and finite
supersoluble groups are infinitely hereditary concerning both the classes
of the FC-groups and hyperfinite groups.
Theorem 10 (L.A. Kurdachenko, J. Otal, I.Ya. Subbotin [10]). Let G be
an infinite locally soluble FC-hypercentral group, D a Dedekind domain
and A an artinian DG-module. If X is a formation, which is infinitely
hereditary concerning the class of FC-hypercentral groups, then A has the
Baer-Shemetkov’s X−DG-decomposition.
The problem of existence of Baer-Shemetkov’s decomposition is
tightly connected to another group theoretical problem, namely the prob-
lem of the existence of complements and supplements to the important
normal subgroups of the group. L.A. Shemetkov also made a significant
contribution to the solution of this problem.
Let G be a group and H be a proper subgroup of G. A proper subgroup
K is said to be a supplement to H in G if G = HK. A supplement K to
H is said to be a complement to H in G if further H ∩K = 〈1〉.
The notion of a complement has been created in the paper [5] of
P. Hall, while the concept of the supplement has been initiated later on
in another Hall’s paper [6]. Note, that for it P. Hall initially used another
term a partial complement.
If H is a normal subgroup of G and H has a complement K, then
it is said that G splits over H and denote this by G = H ⋋ K. If all
complements to H are conjugate, then it is said that G conjugately splits
over H.
Kirichenko, Kurdachenko, Pypka, Subbotin 169
In finite group theory, the problem on the existence of complements
and supplements to normal subgroups has been studied quite deep. Note
here an important theorem of Gaschütz [4]:
If an abelian normal subgroup A of a finite group G has a complement
in a subgroup H and (|G : H|, |A|) = 1, then A has a complement in G.
Moreover, if H is conjugately splits over A, then G is conjugately splits
over A.
The Schur-Zassenhaus’s theorem is deduced from this result. Especially
interesting is the following corollary of Gaschütz’s theorem:
An abelian normal subgroup A of a finite group G has a complement
in G if and only if a Sylow p-subgroup of A has a complement in a Sylow
p-subgroup of G for each prime p.
L.A. Shemetkov has obtained the following generalization of Schur-
Zassenhaus’s and Gaschütz’s theorems.
Theorem 11 (L.A. Shemetkov [15]). Let G be a finite group and π a set
of primes. Let K be a subgroup of G, generated by all π′-elements of G.
If p2 does not divide |K| for every p ∈ π, then K has a complement in G.
Gaschütz’s theorem has two following reductions. Let clG(H) =
{Hg|g ∈ G}, L = CoreG(H) =
⋂
g∈GHg. Clearly A ≤ L, L is a normal
subgroup of G and (|G/L|, |A|) = 1.
Let π be a set of all prime divisors of |A| and σ = Π(G)\π. Denote by
Xσ the class of all finite σ-groups. Clearly Xσ is a formation. In notations
of Gaschütz’s theorem we have G/L ∈ Xσ. The second reduction is
following. We have that A ≤ GXσ , GXσ is a normal subgroup of G and
(|G/GXσ |, |A|) = 1.
This raises the natural question: For which formations does an ana-
log of Gaschütz’s theorem exist? The answer has been obtained by
L.A. Shemetkov in the paper [16]. This result is one of the most general
results on the existence of complements to X-residual in finite groups.
Theorem 12 (L.A. Shemetkov [15]). Let G be a finite group and X be a
local formation.
Let p be a prime such that a Sylow p-subgroup P of GX is abelian. If
S is a Sylow p-subgroup of G, then P has a complement in S.
Suppose that the Sylow q-subgroups of GX are abelian for every prime
divisor q of |G/GX|. Then GX has a complement in G. In particular, if
an X-residual GX of a group G is abelian, then it has a complement in G.
170 On Baer-Shemetkov’s decomposition in modules
The existence of this general result for finite groups was a good
incentive for finding of its analog for infinite groups. However, in the
infinite case, the situation is much more complicated, it is difficult to speak
of an analog of the overall result. However, there is quite a lot of interesting
results for various special cases. A major role in the development of this
subject belongs to D.I. Zaitsev. L.A. Shemetkov was one of those who
drew Zaitsev’s attention on this subject matter. One of the first natural
question here is the existence of supplements to locally nilpotent residual
whenever this residual is abelian. For some natural finiteness conditions
D.I. Zaitsev has obtained the following results.
Theorem 13 (D.I. Zaitsev [22]). Let G be a group and A be an abelian
normal subgroup of G. Suppose that the following conditions hold:
(i) A as ZG-module is artinian;
(ii) CA(G) = 〈1〉;
(iii) G/A is locally nilpotent and G/CG(A) is hypercentral.
Then G splits conjugately over A.
If R is a ring and G is a group, we denote by ω(RG) the augmentation
ideal of the group ring RG.
Theorem 14 (D.I. Zaitsev [21]). Let G be a group and A be an abelian
normal subgroup of G. Suppose that the following conditions hold:
(i) A as ZG-module is noetherian;
(ii) A = A(ω(Z(G/A)));
(iii) G/A is locally nilpotent and G/CG(A) is hypercentral.
Then G splits conjugately over A.
Under some restrictions, some conditions of existence of complements
to the locally nilpotent residual have been obtained by D.J.S. Robinson.
Theorem 15 (D.J.S. Robinson [13]). Let G be a group and A be an
abelian normal subgroup of G. Suppose that the following conditions hold:
(i) A as ZG-module is noetherian;
(ii) G includes a normal subgroup H ≥ A such that H/A is hypercentral;
(iii) A = A(ω(Z(H/A)));
(iv) upper FC-hypercenter of G/CG(A) includes HCG(A)/CG(A).
Then G splits conjugately over A.
Kirichenko, Kurdachenko, Pypka, Subbotin 171
Theorem 16 (D.J.S. Robinson [14]). Let G be a group and A be an
abelian normal subgroup of G. Suppose that the following conditions hold:
(i) A as ZG-module is artinian;
(ii) G includes a normal subgroup H ≥ A such that H/A is hypercentral;
(iii) CA(H) = 〈1〉;
(iv) upper FC-hypercenter of G/CG(A) includes HCG(A)/CG(A).
Then G splits conjugately over A.
Note that Theorem 15 has been proved using only group theory
methods, while the proof of Theorem 16 was based on homology theory. A
purely group-theoretical proof of this Theorem 16 has been presented in the
book [[11], Chapter 17]. The latest general results regarding complements
to the locally nilpotent residual have been obtained in the paper [12].
We say that a normal subgroup L of a group G is said to be G-
hyperfinite if L has an ascending series
〈1〉 = L0 ≤ L1 ≤ ... ≤ Lα ≤ Lα+1 ≤ ... ≤ Lγ = L
of G-invariant subgroups whose factors are finite.
Theorem 17 (L.A. Kurdachenko, J. Otal, I.Ya. Subbotin [12]). Let G be
a group and A be an abelian normal subgroup of G. Suppose that G has
an ascendant subgroup H ≥ A such that H/A is finitely generated and
nilpotent. If A is H-hyperfinite and H-hypereccentric, then G conjugately
splits over A.
Finally we would like to mention that D.I. Zaitsev initiated the search
for the conditions of existance of complements to the locally finite residual.
He obtained the following two results.
Theorem 18 (D.I. Zaitsev [19]). Let G be a group and A be an abelian
normal subgroup of G. Suppose that the following conditions hold:
(i) A as ZG-module is artinian;
(ii) every non-identity G-invariant subgroup of A is infinite;
(iii) G/A is locally soluble and hyperfinite.
Then G splits conjugately over A.
172 On Baer-Shemetkov’s decomposition in modules
Theorem 19 (D.I. Zaitsev [23]). Let G be a group and A be an abelian
normal subgroup of G. Suppose that the following conditions hold:
(i) A as ZG-module is noetherian;
(ii) A does not include proper G-invariant subgroups of finite index;
(iii) G/A is locally soluble and hyperfinite.
Then G splits conjugately over A.
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Contact information
V. V. Kirichenko Department of Geometry, School of Mechan-
ics and Mathematics, Kyiv National Taras
Shevchenko University, Volodymyrska, 64, Kyiv,
01033, Ukraine
E-Mail: vkir@univ.kiev.ua
L .A. Kurdachenko,
A. A. Pypka
Department of Geometry and Algebra, School
of Mechanics and Mathematics, Oles Honchar
Dnipropetrovsk National University, Gagarin
prospect, 72, Dnipropetrovsk, 49010, Ukraine
E-Mail: lkurdachenko@i.ua,
pypka@ua.fm
I. Ya. Subbotin Department of Mathematics and Natural Sci-
ences, College of Letters and Sciences, National
University, 5245 Pacific Concourse Drive, Los
Angeles, CA 90045-6904, USA
E-Mail: isubboti@nu.edu
Received by the editors: 23.04.2013
and in final form 23.04.2013.