Groups with the minimal condition for nonabelian subgroups
Для деяких дуже широких класiв D i B ⊂ D груп автор доводить, що довiльна (неабелева) група G ∊ D (вiдповiдно, G ∊ B) задовольняє умову мiнiмальностi для (неабелевих) пiдгруп тодi i тiльки тодi, коли вона є чернiковською....
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| Цитувати: | Groups with the minimal condition for nonabelian subgroups / N.S. Chernikov // Збірник праць Інституту математики НАН України. — 2006. — Т. 3, № 3. — С. 423-430. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-62772016-05-19T14:09:46Z Groups with the minimal condition for nonabelian subgroups Chernikov, N.S. Геометрія, топологія та їх застосування Для деяких дуже широких класiв D i B ⊂ D груп автор доводить, що довiльна (неабелева) група G ∊ D (вiдповiдно, G ∊ B) задовольняє умову мiнiмальностi для (неабелевих) пiдгруп тодi i тiльки тодi, коли вона є чернiковською. Для некоторых очень широких классов D и B ⊂ D групп автор доказывает, что произвольная (неабелева) группа G ∊ D (соответственно, G ∊ B) удовлетворяет условию минимальности для (неабелевых) подгрупп тогда и только тогда, когда она является черниковской. Для некоторых очень широких классов D и B ⊂ D групп автор доказывает, что произвольная (неабелева) группа G ∊ D (соответственно, G ∊ B) удовлетворяет условию минимальности для (неабелевых) подгрупп тогда и только тогда, когда она является черниковской. For some very wide classes D and B ⊂ D of groups, the author proves that an arbitrary (nonabelian) group G ∊ D (respectively G ∊ B) satisfies the minimal condition for (nonabelian) subgroups iff it is Chernikov. 2006 Article Groups with the minimal condition for nonabelian subgroups / N.S. Chernikov // Збірник праць Інституту математики НАН України. — 2006. — Т. 3, № 3. — С. 423-430. — Бібліогр.: 7 назв. — англ. 1815-2910 http://dspace.nbuv.gov.ua/handle/123456789/6277 519.41/47 en Інститут математики НАН України |
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Геометрія, топологія та їх застосування Геометрія, топологія та їх застосування |
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Геометрія, топологія та їх застосування Геометрія, топологія та їх застосування Chernikov, N.S. Groups with the minimal condition for nonabelian subgroups |
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Для деяких дуже широких класiв D i B ⊂ D груп автор доводить, що довiльна (неабелева) група G ∊ D (вiдповiдно, G ∊ B) задовольняє умову мiнiмальностi для (неабелевих) пiдгруп тодi i тiльки тодi, коли вона є чернiковською. |
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Article |
| author |
Chernikov, N.S. |
| author_facet |
Chernikov, N.S. |
| author_sort |
Chernikov, N.S. |
| title |
Groups with the minimal condition for nonabelian subgroups |
| title_short |
Groups with the minimal condition for nonabelian subgroups |
| title_full |
Groups with the minimal condition for nonabelian subgroups |
| title_fullStr |
Groups with the minimal condition for nonabelian subgroups |
| title_full_unstemmed |
Groups with the minimal condition for nonabelian subgroups |
| title_sort |
groups with the minimal condition for nonabelian subgroups |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| topic_facet |
Геометрія, топологія та їх застосування |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/6277 |
| citation_txt |
Groups with the minimal condition for nonabelian subgroups / N.S. Chernikov // Збірник праць Інституту математики НАН України. — 2006. — Т. 3, № 3. — С. 423-430. — Бібліогр.: 7 назв. — англ. |
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2025-07-02T09:13:08Z |
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2025-07-02T09:13:08Z |
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1836525906797002752 |
| fulltext |
Збiрник праць
Iн-ту математики НАН України
2006, т.3, №3, 423-430
UDC 519.41/47
N. S.Chernikov
Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine
E-mail: chern@imath.kiev.ua
Groups with the minimal condition
for nonabelian subgroups
Для деяких дуже широких класiв D i B ⊂ D груп автор доводить, що
довiльна (неабелева) група G ∈ D (вiдповiдно, G ∈ B) задовольняє
умову мiнiмальностi для (неабелевих) пiдгруп тодi i тiльки тодi, коли
вона є чернiковською.
Для некоторых очень широких классов D и B ⊂ D групп автор дока-
зывает, что произвольная (неабелева) группа G ∈ D (соответственно,
G ∈ B) удовлетворяет условию минимальности для (неабелевых) под-
групп тогда и только тогда, когда она является черниковской.
For some very wide classes D and B ⊂ D of groups, the author proves
that an arbitrary (nonabelian) group G ∈ D (respectively G ∈ B) satisfies
the minimal condition for (nonabelian) subgroups iff it is Chernikov.
Recall that a group G is called Shunkov, if for any its finite
subgroup K every subgroup of the quotient group NG(K)/K,
generated by two its conjugated elements of prime order, is
finite. Recall that a group is called locally graded, if any its
finitely generated subgroup 6= 1 contains a subgroup of finite
index 6= 1 (S.N.Chernikov). The class of all periodic Shunkov
groups is wide and contains, for instance, all binary finite and
2-groups. The class of all locally graded groups is very wide
and contains, for instance, all abelian, locally finite, residually
finite groups. It is easy to see that this class is local and any
group having a series with locally graded factors is locally
c© N. S.Chernikov, 2006
424 Groups with the minimal condition . . .
graded, and any residually (locally graded) group is locally
graded. At the same time, the class of all locally graded
groups contains all groups having a series with locally finite
factors, all RN - (and so all groups of all Kurosh-Chernikov
classes), locally solvable, locally hyperabelian, radical in the
sense of B.I.Plotkin, residually solvable groups.
Below, as usual, π(G) is the set of all primes p for which
the group G has a p-element 6= 1.
Let A be the class of all groups G for which the following
conditions are fulfilled:
(i) If G is not torsion-free, then for any p ∈ π(G) and p-
element g 6= 1 of G and h ∈ CG(gp), 〈g, gh〉 possesses
a subgroup of finite index 6= 1 or 〈g, h〉 is not periodic.
(ii) If G is not periodic, then for any element g of infinite
order of G and h ∈ G, 〈g, gh〉 possesses a subgroup of
finite index 6= 1.
The class A is very wide and contains, for instance, all
periodic Shunkov groups and all locally graded groups.
Let C be the class of all groups G for which (i), with "peri-
odic 6= 1" instead of "not torsion-free" and deleted "or 〈g, h〉
is not periodic", is fulfilled (and (ii) is not necessary fulfilled).
Then A is contained in C, C contains all nonperiodic groups
and the class of all periodic A-groups is just the class of all
periodic C-groups.
Let B (resp. D) be the minimal local class of groups con-
taining A (resp. C) such that any group possessing a series
with B (resp. D)-factors belongs to B (resp. D). Put B0 = A
(resp. D0 = C) and for ordinals β > 0 by induction: if for
some ordinal α, β = α+1, then Bβ (resp. Dβ) be the class of
all groups which have a local system of subgroups possessing
a series with Bα (resp. Dα)-factors, and if there is no such α,
N. S.Chernikov 425
then Bβ =
⋃
α<β
Bα (resp. Dβ =
⋃
α<β
Dα). It is easy to see that
B (resp. D) is the union of classes Bβ (resp. Dβ). It is easy
to show by induction that all Bβ and, at the same time, B
are closed with respect to subgroups.
The known Shunkov’s [1] and S.N.Chernikov’s [2] Theorems
establish that a nonabelian group satisfying the minimal con-
dition for nonabelian subgroups is Chernikov, if it is locally
finite or has a series with finite factors resp. The next Theo-
rem contains them.
Its proof below uses some results [2] and Suchkova-Shunkov
Theorem [3], which asserts that a Shunkov group with the
minimal condition for abelian subgroups is Chernikov.
Theorem. A (nonabelian) group G ∈ D (resp. G ∈ B)
satisfies the minimal condition for (nonabelian) subgroups iff
it is Chernikov.
Note that Ol’shanskii’s nonabelian groups, in which all
proper subgroups are finite (see [4]), satisfy the minimal condi-
tion for subgroups and are non-Chernikov. Thus, in Theorem,
the condition "G ∈ D" is essential. Note that Ol’shanskii’s
nonabelian torsion-free groups, in which all proper subgroups
are cyclic (see [4]), satisfy the minimal condition for non-
abelian subgroups, are Shunkov and non-Chernikov. In par-
ticular, in Theorem, the condition ”G ∈ B” is essential.
Below min and min−ab are the minimal conditions for sub-
groups and nonabelian subgroups resp. Other notations are
standard. (Remark that a group with min is periodic and an
abelian group with min is Chernikov.)
Proof. Sufficiency is obvious.
Necessity. Let G be non-Chernikov.
(a) Reduction to the case when G satisfies min−ab and also
G ∈ A. Let ζ be minimal among all ordinals α, for which Bα
426 Groups with the minimal condition . . .
(resp. Dα) contains a non-(Chernikov or abelian) group (resp.
a non-Chernikov group) with min−ab (resp. min). We may
assume that G ∈ Bζ (resp. G ∈ Dζ). Suppose ζ > 0. Clearly,
for some ordinal ξ, ζ = ξ+1. So G possesses a local system of
subgroups having a series with Bξ (resp. Dξ)-factors. Every
factor is, obviously, Chernikov or abelian (resp. Chernikov).
So G is locally graded. Thus G ∈ B0 (resp. G ∈ D0), which
is a contradiction. So ζ = 0 and G ∈ A (resp. G ∈ C). In the
case of min, G ∈ C and G is, obviously, periodic nonabelian
with min−ab. In particular, G ∈ A. Taking this into account,
we may consider later on only the case of G ∈ A with min−ab.
Since G satisfies min−ab, it contains a non-(Chernikov or
abelian) subgroup L such that any its proper subgroup is
Chernikov or abelian. We may assume that G = L. Then
for every normal subgroup N of G, any proper subgroup of
G/N is Chernikov or abelian.
(b) Show that a subgroup H of G is Chernikov or abelian,
if it has a subgroup K of finite index 6= 1 or if it is almost
solvable. Since K 6= G, K and so H are almost abelian. If
H is almost solvable, then in view of Corollary to Theorem
1 [2], Corollary 2 [2] and Lemmas 1,2 [2], it is Chernikov or
abelian.
(c) Show that G is periodic. Let G have elements g of
infinite order. Since G ∈ A, every 〈g, gh〉 has a proper sub-
group of finite index and so is abelian (see (b)). Then for any
u ∈ G, 〈gh : h ∈ G〉〈u〉 is non-Chernikov solvable and so is
abelian (see (b)). Thus g ∈ Z(G). Let v ∈ G and |〈v〉| < ∞.
Then vg is of infinite order. So v = (vg)g−1 ∈ Z(G). Thus G
is abelian, which is a contradiction.
(d) Show that G/Z(G) is Shunkov. Let K/Z(G) be a finite
subgroup of G/Z(G). In view of Kalužnin’s Theorem (see [5]),
N. S.Chernikov 427
CG(K/Z(G))/CG(K) is abelian. So NG(K)/CG(K) is almost
abelian.
If K/Z(G) 6= 1, then CG(K) 6= G and so CG(K) is al-
most abelian. Thus NG(K) is almost solvable. So NG(K)
is Chernikov or abelian (see (b)). Clearly, NG(K)/Z(G) =
NG/Z(G)(K/Z(G)). Consequently, the quotient group
NG/Z(G)(K/Z(G))/(K/Z(G)) is Chernikov or abelian. There-
fore any two its elements of prime order generate a finite sub-
group.
Let K/Z(G) = 1 and R/Z(G) be a subgroup of G/Z(G)
generated by two its conjugated element of some prime or-
der p. Obviously, because of Z(G) is periodic (see (c)), for
some p-elementg ∈ G such that gp ∈ Z(G) and some h ∈ G,
R = 〈g, gh〉Z(G). Clearly, R/Z(G) is isomorphic to a quotient
group of the group 〈g, gh〉/〈gp〉. Since G ∈ A, 〈g, gh〉 has a
subgroup of finite index 6= 1. So, with regard to (b), 〈g, gh〉
is finite. Therefore R/Z(G) is finite.
(e) Show that G/Z(G) has an abelian non-Chernikov non-
normal maximal subgroup A/Z(G) such that
(0.10)
A/Z(G) ∩ (A/Z(G))g = 1, g ∈ (G/Z(G))\(A/Z(G)).
If all proper subgroups of G/Z(G) are Chernikov, then it
satisfies min. So because of G/Z(G) is Shunkov (see (d)),
by Suchkova-Shunkov Theorem [3] it is Chernikov. So G
is almost solvable, which is a contradiction (see (b)). So
some maximal abelian subgroup A/Z(G) of G/Z(G) is non-
Chernikov. An arbitrary proper subgroup H ⊇ A of G is non-
Chernikov. So it is abelian. Therefore H/Z(G) = A/Z(G)
and H = A. Thus A is an abelian maximal subgroup of G.
If A is normal in G, then |G : A| is prime and G is solv-
able, which is a contradiction (see (b)). Consequently, for
any g ∈ G\A, G = 〈A, Ag〉. Then A ∩ Ag ⊆ Z(G). But,
428 Groups with the minimal condition . . .
clearly, Z(G) ⊆ A, Ag. Thus A ∩ Ag = Z(G). Therefore (1)
is valid.
(f) Show that A/Z(G) has some element a of odd prime
order. Suppose that this is not the case. Since A/Z(G) is pe-
riodic (see (c)) and neither cyclic nor quasicyclic, it has some
elements b and c 6= b of order 2. Let h ∈ (G/Z(G))\(A/Z(G)).
Then 〈b, ch〉 = 〈bch〉⋋ 〈b〉 = 〈bch〉⋋ 〈ch〉 and |〈bch〉| < ∞ (see
(c)). If |〈bch〉| is odd, then for some s ∈ 〈bch〉, b = chs.
Since A/Z(G) is abelian and b, c ∈ A/Z(G), hs /∈ A/Z(G).
But b ∈ A/Z(G) ∩ (A/Z(G))hs, which is a contradiction (see
(1)). So 〈bch〉 contains some element w of order 2. But then
w ∈ CG/Z(G)(b) ∩ CG/Z(G)(c
h) = A/Z(G) ∩ (A/Z(G))h, which
is a contradiction.
(g) Final contradiction. Let a be from (f). Since G/Z(G)
is Shunkov (see (d)), for any h ∈ G/Z(G), |〈a, ah〉| < ∞. So
with regard to (1) by Sozutov-Shunkov Theorem [6], for some
normal subgroup N/Z(G) of G/Z(G),
G/Z(G) = (A/Z(G))(N/Z(G)) and A/Z(G) ∩ N/Z(G) = 1.
Since N 6= G and G/N is abelian, G is almost solvable, which
is a contradiction (see (b)).
The following new proposition is contained in Theorem.
Proposition 1. Let G be a nonabelian group. Assume
that G is locally graded or periodic Shunkov. Then G satisfies
min−ab iff it is Chernikov.
In view of Mal’cev Theorem (see Theorem 4.2 [7]), a linear
group over a field is locally residually finite. Further, for a
commutative and associative ring R with 1 and any finitely
generated unital module M over R, AutR(M) is hyperabelian-
by-residually (linear over fields) (Theorem 13.5 [7]). Conse-
quently, AutR(M) is locally graded. Hence follows that any
N. S.Chernikov 429
GLn(R) is locally graded. Therefore, in virtue of Theorem,
the following proposition is valid.
Proposition 2. A (nonabelian) group G ⊆ AutR(M) or
G ⊆ GLn(R) satisfies min (resp. min−ab) iff it is Chernikov.
Finally, let E be the class of all groups G for which the
following conditions are fulfilled:
(i) If G is not torsion-free, then for any p ∈ π(G) and p-
element g 6= 1 of G and h ∈ CG(gp), 〈g, gh〉 possesses
a B-homomorphic image 6= 1 or 〈g, h〉 is not periodic.
(ii) If G is not periodic, then for any element g of infi-
nite order of G and h ∈ G, 〈g, gh〉 possesses a B-
homomorphic image 6= 1.
Let F be the class of all groups G for which the following
condition is fulfilled:
If G is periodic 6= 1, then for any p ∈ π(G) and p-element
g 6= 1 of G and h ∈ CG(gp), 〈g, gh〉 possesses a
D-homomorphic image 6= 1.
Proposition 3. A (nonabelian) group G ∈ F (resp. G ∈
E) satisfies min (resp. min−ab) iff it is Chernikov.
Proof. Necessity. A corresponding homomorphic image
of 〈g, gh〉 is generated by two elements. In the case when it
is not abelian, by Theorem, it is finite. In the case when it is
abelian, it has a subgroup of finite index 6= 1. Consequently,
〈g, gh〉 possesses a subgroup of finite index 6= 1. So G ∈ C
(resp. G ∈ A). Therefore by Theorem, G is Chernikov.
Sufficiency is obvious.
430
References
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[2] Chernikov S.N. Groups with the minimal condition for nonabelian sub-
groups // Groups with restrictions for subgroups. – Kyiv: Naukova
dumka, 1971. – P. 96–106. (in Russian).
[3] Suchkova N.G., Shunkov V.P. On groups with the minimal condition for
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Russian).
[4] Ol’shanskii A.Yu. Geometry of defining relations in groups. – Moscow:
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