Algebra and Discrete Mathematics SURVEY ARTICLE
Volume 22 (2016). Number 2, pp. 184–200
© Journal “Algebra and Discrete Mathematics”
Groups satisfying certain rank conditions
Martyn R. Dixon, Leonid A. Kurdachenko,
Aleksandr A. Pypka, Igor Ya. Subbotin
To Alexander Olshanskii, for his 70th birthday
Abstract. This is a survey of a number of recent results
concerned with groups whose subgroups satisfy certain rank condi-
tions.
Introduction
The concept of “rank” in group theory has its roots in vector space
theory, where the analogous notion of dimension is so important. Groups
also are associated with numerical characteristics, some of which in one
way or another are analogs of the concept of the dimension of a vector
space. These numerical invariants are not universal, and since they are
introduced and employed in specific classes of groups, their nature is local.
One of the earliest numerical characteristics is as follows.
Let G be a finitely generated group. Denote by d(G) the smallest
number of generators of the group G.
Here have a complete analogy with vector spaces: the dimension of a
vector space is precisely the number of elements in a minimal system of
generators. However, this analogy ends at this point: a group may have
minimum sets of generators with a different number of elements. Besides,
finitely generated groups can have non-finitely generated subgroups. More-
over, a finitely generated group G, whose subgroups are finitely generated
(that is the group satisfying the maximal condition for all subgroups), can
2010 MSC: Primary: 20E15; Secondary: 20E25, 20F50.
Key words and phrases: finite rank, torsion-free rank, section p-rank.
M. Dixon, L. Kurdachenko, A. Pypka, I. Subbotin 185
include a subgroup H such that d(H) > d(G). Thus, unlike dimension,
the property of homogeneity is not observed in groups.
Now we consider other numerical invariants, in varying degrees preserv-
ing the analogy with the notion of dimension. One of the first extensions
of the concept of dimension was the concept of the R-rank of a module A
over a (commutative) ring R. Since every abelian group is a module over
the ring Z of integers, the concept of Z-rank (in the theory of abelian
groups we use the term 0-rank) has been resourcefully used in the theory
of abelian groups. Starting from the Z-rank, first we define the 0-rank, or
torsion-free rank, of a group. In the paper [48], A.I. Maltsev introduced
the class of soluble A1-groups, a class consisting of those groups having a
finite subnormal series whose factors are periodic abelian or torsion-free
and locally cyclic. These groups also possess finite subnormal series, the
factors of which are periodic or infinite cyclic and it was observed that
the number of infinite cyclic factors is invariant.
If G is any group which has an ascending series whose factors are
either infinite cyclic or periodic and if the number of infinite cyclic factors
is finite then the group G is said to have finite 0-rank. In this case the
0-rank of the group G is the number of infinite cyclic factor-groups in
the series and is denoted by r0(G). It is not hard to see that r0(G) is
an invariant of G. If the number of infinite cyclic factors is infinite, then
we say that G has infinite 0-rank. If G has no such ascending series the
0-rank is undefined.
For an abelian group G, the cardinality of a maximal Z-independent
subset of G (i.e. the Z-rank) is precisely the torsion-free rank of G. Of
course, the 0-rank of a periodic group is 0.
For periodic abelian groups a very important role is played by the
p-rank. It is defined as the dimension of the lower layer of its Sylow
p-subgroup. One characterization of the p-rank introduces us to the
following concept.
Let p be a prime. If every elementary abelian p-section of G is finite
of order at most pr and there is an elementary abelian p-section of order
precisely pr, then G is said to have finite section p-rank r, denoted by
srp(G) = r.
The ranks described above are connected with another important
characteristic of a group, namely its special rank. It is based on the
following property of dimension. If A is a vector space of finite dimension
k over a field F and B is a subspace of A, then it is well-known that
B is finite dimensional and that the dimension of B is at most k. We
say that a group G is of finite special rank r(G) = r if every finitely
186 Groups satisfying certain rank conditions
generated subgroup can be generated by at most r elements, and r is the
least integer with this property. This concept was introduced for arbitrary
groups by A.I. Maltsev [47]. Very often the term “rank” is used instead of
the “special rank”, which is also sometimes called the “Prüfer rank”. The
study of groups of finite ranks has been of central importance in infinite
group theory.
In this paper we first give a brief history of the theory of groups
satisfying the various rank conditions. Many classical theorems have been
devoted to this topic. For more extensive papers on this topic the reader
should consult [26] or [59]. In Section 2 we consider groups whose proper
subgroups typically satisfy some rank conditions; this involves more recent
results as well as certain of the classical results. In Section 3 we discuss
recent work concerned with groups whose subgroups of infinite special
rank satisfy some type of normality condition. In Section 4 we discuss
results concerning groups whose subgroups are extensions of groups of
finite special rank. Our notation is generally standard and can be found
in [58].
1. Groups satisfying rank conditions
It is not our intention here to dwell too much on the history of groups
satisfying rank conditions, but it is helpful to put some of the later results
in context by briefly discussing some of the main results.
D.I. Zaitsev discussed the class of groups of finite 0-rank in a number
of papers, including [67–69], where he demonstrated the usefulness of the
concept. In the paper [66] D.I. Zaitsev discussed a special class of groups
for which the 0-rank is finite, namely the class of polyrational groups,
where a group is polyrational if it has a subnormal series whose factors are
torsion-free locally cyclic groups, and hence are subgroups of the rationals.
For groups of finite 0-rank, probably the most general theorem obtained
so far occurs in [25] where the following theorem is proved.
We recall that a group is called generalized radical if it has an ascending
series whose factors are either locally nilpotent or locally finite.
Theorem 1. Let G be a locally generalized radical group of finite 0-rank.
Then G has normal subgroups T 6 L 6 K 6 S 6 G such that
(i) T is locally finite and G/T is soluble-by-finite of finite special rank;
(ii) L/T is a torsion-free nilpotent group;
(iii) K/L is a finitely generated torsion-free abelian group;
(iv) G/K is finite and S/T is the soluble radical of G/T .
M. Dixon, L. Kurdachenko, A. Pypka, I. Subbotin 187
Moreover, if r0(G) = r, then there are the functions f1, f2 such that
|G/K| 6 f1(r) and dl(S/T ) 6 f2(r).
A key role, in the proof of this and many other results mentioned
here, is played by the following beautiful theorem due to A.I. Maltsev
[48, Theorem 5].
Theorem 2. Let G be a torsion-free locally nilpotent group. Suppose that
every abelian subgroup of G has finite 0-rank. Then G is a nilpotent group
of finite 0-rank. Moreover, if A is a maximal normal abelian subgroup in
G and r0(A) = k, then r0(G) 6 12k(k + 1) and G is nilpotent of class at
most 2k.
We note also the following local property of 0-rank, which has been
proved in [25].
Theorem 3. Let G be a group and suppose that G satisfies the following
conditions:
(i) for every finitely generated subgroup L of a group G the factor-group
L/Tor(L) is a locally generalized radical group;
(ii) there is a positive integer r such that r0(L) 6 r for every finitely
generated subgroup L.
Then G/Tor(G) includes a normal soluble subgroup D/Tor(G) of finite
index. Moreover, G has finite 0-rank r and there is a function f3 such
that |G/D| 6 f3(r).
For locally finite p-groups, G, it is well-known that the finiteness of
srp(G) is equivalent to G being Chernikov, which itself is well-known to
be equivalent to the finiteness of the special rank of G. Moreover, in [5] it
has been proved that for a locally finite p-group G we have srp(G) = r(G).
This means more can be said concerning locally finite groups with finite
section p-rank for all primes p. A result of V.V. Belyaev [6] shows that
such groups are almost locally soluble and somewhat more of the structure
of such groups can then be deduced (see [43], for example). For torsion-
free locally nilpotent groups, G, with finite section p-rank r (for some
prime p) it is proved in [5] that G is nilpotent of class at most r and
srp(G) = r0(G).
For groups in general the following theorem is known (see [25]).
Theorem 4. Let G be a locally generalized radical group of finite sec-
tion p-rank rp for some prime p. Then G has finite 0-rank at most 2rp.
Furthermore, G has normal subgroups T 6 L 6 K 6 S 6 G such that
188 Groups satisfying certain rank conditions
(i) T is a locally finite group whose Sylow p-subgroups are Chernikov;
(ii) L/T is a torsion-free nilpotent group;
(iii) K/L is a finitely generated torsion-free abelian group;
(iv) G/K is finite and S/T is the soluble radical of G/T .
Moreover, there are the functions f4, f5 such that |G/K| 6 f4(rp),
dl(S/T ) 6 f5(rp).
Certainly, groups of finite special rank have finite section p-rank,
for each prime p, so special cases of Theorems 1 and 4 hold for locally
generalized radical groups of finite special rank. In these special cases
more is known. When the main theorem of [6] is combined with a result
of M.I. Kargapolov [41], for example, the following result can be obtained.
Theorem 5. Let G be a locally generalized radical group of finite special
rank r. Then its locally nilpotent radical L is hypercentral and G/L includes
a normal abelian subgroup K/L such that G/K is finite. In particular, G
is generalized radical, even almost hyperabelian. Moreover, Tor(L) is a
direct product of its Chernikov Sylow p-subgroups, L/Tor(L) is nilpotent,
K/Tor(L) has finite 0-rank at most r. In particular, G has finite 0-rank,
moreover, r0(G) 6 r.
Recall that a group G is locally graded if every finitely generated
non-trivial subgroup of G has a finite non-trivial image. Denote by X
the class of groups, obtained from the class of periodic locally graded
groups by using of the formation of local systems, subcartesian products
and both ascending and descending normal series. In [11], N.S. Chernikov
proved that the groups of finite special rank from this class are almost
locally soluble.
In the papers [17,21] groups that are residually of finite special rank
have been considered. The main result of these papers are as follows.
Theorem 6. Let G be a group and suppose that G has a family of normal
subgroups S such that ⋂S = 〈1〉 and G/H is a locally (soluble-by-finite)
group of special rank at most r for each H ∈ S, where r is a fixed positive
integer. Then either G is locally (soluble-by-finite) or G includes a non-
abelian free subgroup.
It is known that not every residually finite p-group is locally finite.
However,
Theorem 7. Let G be a periodic group and suppose that G has a family
of normal subgroups S such that ⋂S = 〈1〉 and G/H is a locally finite
M. Dixon, L. Kurdachenko, A. Pypka, I. Subbotin 189
group of special rank at most r for each H ∈ S, where r is a fixed positive
integer. Then G is locally finite.
Theorem 8. Let G be a finitely generated group and suppose that G has
a family of normal subgroups S such that ⋂S = 〈1〉 and G/H is a soluble
group of special rank at most r for each H ∈ S, where r is a fixed positive
integer. Then G includes a normal abelian-by-nilpotent subgroup Q such
that G/Q is subdirect product of finitely many linear groups. If G includes
no non-abelian free subgroups then G is nilpotent-by-abelian-by-finite.
Theorem 9. Let p be a prime and G be a free group with a countable set
of free generators. Then G has a descending series
G = K0 > K1 > . . . > Kn > Kn+1 > . . .
⋂
j∈N
Kj = 〈1〉
of normal subgroups such that G/Kn is a finite p-group of special rank 9
for all n ∈ N.
Theorem 6 generalizes the now well-known result of A. Lubotzky and
A. Mann [46] and N.S. Chernikov [11] that a residually finite group of
finite special rank is almost locally soluble.
When one leaves the class of generalized soluble groups, the theory of
groups of finite special rank is beset with difficulties. It is well-known that
A.Yu. Olshanskii [53] has constructed infinite 2-generator simple groups
with all proper subgroups cyclic of order p, for large enough primes p.
Such groups are of rank 2 and have finite section p-rank 1.
2. Groups in which some system of subgroups have finite
rank
In this survey we are interested in groups in which certain proper
subgroups satisfy one of the rank conditions described above. The theory
of groups all of whose abelian subgroups satisfy some finiteness condition
on their ranks is well documented. In particular, in [42], M.I. Kargapolov
showed that a soluble group all of whose abelian subgroups have finite
special rank itself has finite special rank and, in [49], Yu.I. Merzljakov
showed that a locally soluble group in which all abelian subgroups have
bounded special rank also has finite special rank. On the other hand,
Yu.I. Merzljakov [50] showed that there exists a non-periodic locally poly-
cyclic group, having infinite special rank, in which all abelian subgroups
have finite (not bounded) special rank. However for periodic groups the
190 Groups satisfying certain rank conditions
situation is better. V.P. Shunkov [63] showed that a locally finite group
all of whose abelian subgroups have finite special rank itself has finite
special rank. A result of V.V. Belyaev [6] shows that a locally finite group
whose abelian subgroups have finite section p-rank for all primes p itself
has finite section p-rank for all primes p.
Next we recall that a group is said to have finite abelian subgroup rank
if the rank of every abelian p-subgroup is finite for all primes p and if the
0-rank of every abelian subgroup is finite. The class of groups with finite
abelian subgroup rank is not closed under taking homomorphic images,
so that it is sometimes more useful to consider groups with finite abelian
section rank (we say that G is a group of finite abelian section rank, if
every elementary abelian section of G is finite). Such groups have been
discussed in [59]. Of course if G is a group with all abelian subgroups of
finite special rank then G has finite abelian subgroup rank. Furthermore,
if G is a group with finite section p-rank for all primes p, then G has
finite abelian subgroup rank and it is remarkable that these two classes
of groups quite often coincide. The main result of a paper of R. Baer and
H. Heineken [4] gives the structure of radical groups with finite abelian
subgroup rank and shows that such groups have finite section p-rank
for all primes p; [6] shows that locally finite groups with finite abelian
subgroup rank have finite section p-rank, for all primes p.
In [15] a partial generalization of the theorem of N.S. Chernikov was
obtained. It was proved in [15] that if G is a locally (soluble-by-finite)
group in which every locally soluble subgroup of G has finite special rank,
then G has finite special rank and is almost locally soluble. Furthermore if
G is a locally (soluble-by-finite) group with all abelian periodic subgroups
of finite special rank and all torsion-free abelian subgroups of bounded
special rank, then G has finite special rank and is almost locally soluble
(see [15]).
A group has bounded torsion-free abelian subgroup rank if there is
a bound on the 0-ranks of the torsion-free abelian subgroups. If N is a
soluble group with finite abelian subgroup rank and bounded torsion-free
rank r0, then [59, Theorem 10(iii)] shows that there exists an integer d
depending on r0 only such that N (d) is periodic. Then, by [59, Theorem
10(iv)], there is an integer h(r0) such that each torsion-free abelian factor
of N has 0-rank at most h(r0). One consequence of these ideas is the
theorem, from [20], that ifG is locally (soluble-by-finite) with finite abelian
subgroup rank and bounded torsion-free abelian subgroup rank, then G
is almost locally soluble.
M. Dixon, L. Kurdachenko, A. Pypka, I. Subbotin 191
If G is a free group then all abelian, and indeed all locally soluble,
subgroups of G have special rank 1. Thus a locally graded group with
all locally soluble subgroups of finite special rank need not have finite
special rank. It seems to be unknown whether the following question has
an affirmative answer or not: if G is a periodic locally graded group and
all locally soluble subgroups of G have finite special rank then does G
have finite special rank? The structure of locally graded groups with all
proper subgroups of finite special rank also seems to be unknown.
We conclude this section with the remarkable examples that can
be constructed using the following theorem of A.Yu. Olshanskii (see
[54, Theorem 35.1]).
Theorem 10. Let {Gλ|λ ∈ Λ} be a finite or countable set of non-trivial
finite or countably infinite groups without involutions. Suppose |Λ| > 2
and that n is a sufficiently large odd number (at least n > 1075). Suppose
Gλ ∩ Gµ = 〈1〉 for λ 6= µ, λ, µ ∈ Λ. Then there is a countable simple
group G = OG(Gλ|λ ∈ Λ), containing a copy of G for all λ ∈ Λ with the
following properties:
(i) if x, y ∈ G and x ∈ Gλ\〈1〉, y 6∈ Gλ for some λ ∈ Λ, then G = 〈x, y〉;
(ii) every proper subgroup of G is either a cyclic group of order dividing
n or is contained in some subgroup conjugate to some Gλ.
Hence, these groups of A.Yu. Olshanskii’s are 2-generator and have
subgroups which are restricted by the choice of the constituent groups
Gλ. An application of this theorem allows us to construct the following
examples.
Example 1.
(i) There is a 2-generator group G of infinite special rank, all of whose
proper subgroups have finite special rank.
(ii) There is a periodic 2-generator group G of infinite special rank with
all proper subgroups abelian of finite special rank.
(iii) There is a group of infinite special rank, which is not finitely gener-
ated, with all proper subgroups of finite special rank.
(iv) Let p be a prime such that p > 1075. Then there exist an uncountable
p-group G of finite special rank.
3. Groups with proper subgroups of infinite rank
Many recent papers have been concerned with groups of infinite spe-
cial rank whose proper subgroups of infinite special rank satisfy some
192 Groups satisfying certain rank conditions
property; typical such properties include nilpotency, normality and their
generalizations and we here give a brief summary of these ideas. For
example, if the non-abelian subgroups of a group G have finite special
rank when does it follow that all the subgroups of G have finite special
rank, and what conclusion can then be drawn concerning G?
Let P be a group theoretical property or class of groups. As usual, we
shall say that G is a P-group, or that G ∈ P, if G has the property P or
belongs to the class P. We let P∗ denote the class of groups G in which
every proper subgroup of G is a P-group or a subgroup of finite special
rank. As usual we let N, S denote the classes of nilpotent and soluble
groups respectively, with a subscript c denoting nilpotent of class at most
c (respectively of derived length at most c). We also let LN denote the
class of locally nilpotent groups.
Suppose G is in the class N∗1, so all proper subgroups of G are abelian
or of finite special rank. Clearly groups which themselves are abelian, or
are of finite special rank, fall into this class. It was proved in [19] that if c
is a positive integer and if G ∈ X∩N∗c , then either G ∈ Nc or G has finite
special rank. In either case G is almost locally soluble. The paper [19]
contains other results of this nature. For example, if G ∈ X∩ (LN)∗, then
either G ∈ LN, or G has finite special rank. In either case G is almost
locally soluble.
Of course there is no corresponding result if we replace Nc by N
here since the Heineken-Mohamed groups [39] are locally nilpotent, non-
nilpotent and of infinite special rank, but have all proper subgroups
nilpotent. However the existence of such groups is essentially the only
reason that we cannot replace Nc by N in the results mentioned above.
At this point it is natural to ask if one can obtain similar results for
groups in the class S∗d. One potential obstacle to such a classification
is the existence of locally finite simple groups of infinite special rank
in which every proper subgroup is finite or metabelian. For example,
the group PSL(2,F) has this property if F is a locally finite field with
no proper infinite subfields (see [56]). In order to settle this question
information concerning soluble groups with all proper subgroups soluble
of derived length d is required. The results mentioned above require a
well-known result of O.Yu. Schmidt [61] which states that a finite group
with all proper subgroups nilpotent is soluble. For groups with all proper
subgroups soluble of bounded derived length, a slight extension of a
theorem of D.I. Zaitsev [65], that an infinite soluble group of derived
length exactly d has a proper subgroup of derived length precisely d, is
required.
M. Dixon, L. Kurdachenko, A. Pypka, I. Subbotin 193
Using this result the following theorem can be obtained (see [16]).
Theorem 11. Let G ∈ X ∩ (LS)∗. Then either
(i) G is locally soluble or
(ii) G has finite special rank or
(iii) G is isomorphic to one of SL(2,F), PSL(2,F) or Sz(F) for some
infinite locally finite field F in which every proper subfield is finite.
Here Sz(F) denotes the Suzuki group over the field F. We note that
each of the groups in (iii) actually is in the class under consideration.
A number of very recent papers concerning groups of infinite special
rank whose proper subgroups of infinite special rank satisfy some normality
condition have extended our knowledge in this area (see [12,13,28–37], for
example). As is well-known a group in which every subgroup is normal
is called a Dedekind group, and such finite groups were classified by
R. Dedekind in [14]. What then happens if every subgroup of infinite
special rank is a normal subgroup? In [38] a number of results were
obtained in this direction. First the authors showed in [38, Theorem 2.3]
that a locally graded group with all subgroups Chernikov or normal is
itself either Chernikov or a Dedekind group, a result having a number
of other consequences. They show in [38, Corollary 2.5] that if G is a
locally graded group with all proper subgroups soluble of finite abelian
section rank, then G is also soluble with finite abelian section rank. In
[38, Theorem 2.7] the authors obtain the following theorem.
Theorem 12. Let G be a locally graded group. Then every non-normal
subgroup of G is soluble with finite abelian section rank if and only if G
satisfies one of the following conditions:
(i) G is a Dedekind group;
(ii) G is soluble with finite abelian section rank;
(iii) G has finite abelian section rank and contains a finite normal mini-
mal non-soluble subgroup N such that G/N is a Dedekind group. In
particular, [G,G] is finite.
This result is used to obtain precise conditions for the structure of a
locally graded group with all non-normal subgroups polycyclic.
In general the problem of deciding the structure of a locally graded
group whose subgroups are normal or of finite special rank seems more
complicated, again because our knowledge of locally graded groups with
all proper subgroups of finite special rank is limited.
The papers [44,45] and [27] start to rectify this. A well-known theorem
of Roseblade [60] asserts that if G is a group in which every subgroup is
194 Groups satisfying certain rank conditions
subnormal of defect at most d, then G is a nilpotent group (with class
dependent on d). In [27] the authors study the class of X-groups all of
whose proper subgroups are of finite special rank or subnormal of defect
at most d. It is proved there that if G is an X-group of infinite special
rank in which every proper subgroup of infinite special rank is subnormal
of defect at most d, then G is nilpotent of nilpotency class bounded by
a function of d. In particular, if every subgroup of infinite special rank
is normal then G is a Dedekind group, so is nilpotent of class at most 2.
This result can also be deduced from the results of paper [44].
Earlier in [45] it was proved that if G is a soluble group of infinite
special rank whose proper subgroups of infinite special rank are subnormal,
then G is a Baer group. This result was taken a step further in [44] where
it is shown that a locally (soluble-by-finite) group of infinite special rank,
all of whose proper subgroups of infinite special rank are subnormal, is in
any case soluble therefore generalizing a famous result of W. Möhres [51],
who showed that a group in which all subgroups are subnormal is soluble.
Indeed, for torsion-free locally (soluble-by-finite) groups G of infinite
special rank whose proper subgroups of infinite special rank are subnormal,
the conclusion is that G is nilpotent [44]. In this case, however, an example
is exhibited of a metabelian locally nilpotent group G of infinite special
rank, whose torsion subgroup T has finite special rank and which contains
a non-subnormal subgroup. It turns out though that every subgroup of
infinite special rank is subnormal.
There are other natural generalizations of normality such as per-
mutability. It is well-known that O. Ore [55] proved that in a finite group
G all permutable subgroups of G are subnormal in G. Permutable sub-
groups have also been called quasinormal subgroups by some authors, and
during his investigations of permutable subgroups, S.E. Stonehewer [64]
proved that all permutable subgroups of a group G are ascendant in G.
Groups with all subgroups permutable, so called quasihamiltonian
groups, were classified quite precisely by K. Iwasawa [40]. The following
theorem summarizes some of the main results and proofs can be found
in [62], for example.
Theorem 13. Let G be a non-abelian quasihamiltonian group. Then
(i) G is locally nilpotent;
(ii) G is metabelian;
(iii) if Tor(G) is the torsion subgroup of G and if G is not periodic then
Tor(G) is abelian and G/Tor(G) is a torsion-free abelian group of
special rank 1.
M. Dixon, L. Kurdachenko, A. Pypka, I. Subbotin 195
Groups all of whose subgroups of infinite special rank are permutable
were studied in [24] where it was shown that, for the class X, if G is
an X-group having infinite special rank and every subgroup of infinite
special rank is permutable then G is quasihamiltonian. This result was
generalized in a certain subclass of X in [32], where it was also proved
that if G is a periodic locally graded group of infinite special rank and if
all subgroups of infinite special rank are modular, then G has modular
subgroup lattice.
A group G is called metahamiltonian if the non-normal subgroups are
abelian. Recently groups whose non-abelian subgroups of infinite special
rank are normal have been studied (see [34]) and the following theorem
obtained.
Theorem 14. Let G be a locally (soluble-by-finite) group of infinite
special rank in which every non-abelian subgroup of infinite special rank
is normal. Then G is metahamiltonian.
4. Groups whose subgroups are (finite rank)-by-soluble
Now let R denote the class of groups of finite special rank. Finite
groups with all subgroups nilpotent were studied in [61] and shown to be
soluble, but again the examples of A.Yu. Olshanskii [53] show that this is
not true in general. In a series of papers B. Bruno [7–9] and B. Bruno
and R. Phillips [10] showed that a locally graded, non-perfect group,
which is not nilpotent-by-finite but all of whose proper subgroups are
nilpotent-by-finite must be periodic. This continued a line of research
concerned with the notion of minimal non-Y-groups, where Y is some class
of groups. A minimal non-Y-group is a group with all proper subgroups
in Y, but the group itself is not. In [57], locally graded groups all of whose
proper subgroups are Chernikov-by-nilpotent and locally graded groups
with all proper subgroups nilpotent-by-Chernikov were considered and,
in a follow-up paper, F. Napolitani and E. Pegoraro [52] proved that a
locally graded group with all proper subgroups nilpotent-by-Chernikov
is either nilpotent-by-Chernikov or is a perfect locally finite p-group in
which all proper subgroups are nilpotent. In [2] it was proved that a
locally nilpotent p-group in which every proper subgroup is nilpotent-by-
Chernikov is itself nilpotent-by-Chernikov. In particular this meant that
the following theorem is true.
Theorem 15. Let G be a locally graded group all of whose proper sub-
groups is nilpotent-by-Chernikov. Then G is nilpotent-by-Chernikov and
hence is soluble.
196 Groups satisfying certain rank conditions
This result has inspired many generalizations. For example, in [22],
groups with all proper subgroups (finite rank)-by-nilpotent were consid-
ered. The following result was obtained.
Theorem 16. Let G be a locally (soluble-by-finite) group and suppose
that every proper subgroup of G belongs to RN.
(i) If G is a p-group for some prime p then either G ∈ RN or
G/[G,G] ∼= Cp∞ and every proper subgroup of G is nilpotent.
(ii) If G is not a p-group then G ∈ RN.
When a bound is placed on the nilpotency classes things are more
straightforward (see [18]). In this case, if G is a locally (soluble-by-finite)
group and every proper subgroup of G belongs to the class RNc, then
G ∈ RNc.
This theorem was generalized in [23].
Theorem 17. Let G be a locally (soluble-by-finite) group and suppose
that every proper subgroup of G belongs to SR. Then either
(i) G is locally soluble, or
(ii) G is soluble-by-(finite rank) and almost locally soluble, or
(iii) G is soluble-by-PSL(2,F), or
(iv) G is soluble-by-Sz(F),
where F is an infinite locally finite field with no infinite proper subfields.
There are a number of other results of the types discussed here,
including [1, 3], for example.
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Contact information
M. R. Dixon Department of Mathematics, University of Al-
abama, Tuscaloosa, AL 35487-0350, USA
E-Mail(s): mdixon@ua.edu
L. A. Kurdachenko,
A. A. Pypka
Department of Geometry and Algebra, Faculty
of mechanics and mathematics, Oles Honchar
Dnipropetrovsk National University, Gagarin
prospect, 72, Dnipro, 49010, Ukraine
E-Mail(s): 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, LA,
CA 90045, USA
E-Mail(s): isubboti@nu.edu
Received by the editors: 09.03.2016.