Minimal non-PC-groups
The purpose of this paper is to prove that a non-perfect group G is a minimal non-PC-group if and only if it is a minimal non-FC-group. It is shown that a perfect locally graded minimal non-PC-group is an indecomposable countable locally finite p-group.
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nasplib_isofts_kiev_ua-123456789-1533412025-02-23T18:30:04Z Minimal non-PC-groups Artemovych, O.D. The purpose of this paper is to prove that a non-perfect group G is a minimal non-PC-group if and only if it is a minimal non-FC-group. It is shown that a perfect locally graded minimal non-PC-group is an indecomposable countable locally finite p-group. 2014 Article Minimal non-PC-groups / O.D. Artemovych // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 1–7. — Бібліогр.: 13 назв. — англ. 1726-3255 2010 MSC:20F24, 20E45. https://nasplib.isofts.kiev.ua/handle/123456789/153341 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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The purpose of this paper is to prove that a non-perfect group G is a minimal non-PC-group if and only if it is a minimal non-FC-group. It is shown that a perfect locally graded minimal non-PC-group is an indecomposable countable locally finite p-group. |
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Artemovych, O.D. |
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Artemovych, O.D. Minimal non-PC-groups Algebra and Discrete Mathematics |
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Artemovych, O.D. |
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Artemovych, O.D. |
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Minimal non-PC-groups |
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Minimal non-PC-groups |
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Minimal non-PC-groups |
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Minimal non-PC-groups |
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Minimal non-PC-groups |
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minimal non-pc-groups |
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Інститут прикладної математики і механіки НАН України |
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Minimal non-PC-groups / O.D. Artemovych // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 1–7. — Бібліогр.: 13 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT artemovychod minimalnonpcgroups |
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2025-11-24T10:28:46Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 18 (2014). Number 1, pp. 1 – 7
© Journal “Algebra and Discrete Mathematics”
Minimal non-PC-groups
Orest D. Artemovych
Communicated by L. A. Kurdachenko
Abstract. The purpose of this paper is to prove that a
non-perfect group G is a minimal non-PC-group if and only if it is
a minimal non-FC-group. It is shown that a perfect locally graded
minimal non-PC-group is an indecomposable countable locally finite
p-group.
1. Introduction
A group G is called a PC-group if the quotient group G/CG(xG) is
polycyclic-by-finite for all x ∈ G [1]. The class of PC-groups is closed
with respect to subgroups, quotients and direct products of its members
and contains FC-groups (that is groups with finite conjugacy classes).
Recall that a group G is called non-perfect if the derived subgroup G′ is
proper in G, and is called perfect otherwise. Moreover, a group is locally
graded if its every finitely generated subgroup contains a proper subgroup
of finite index [2]. Recall also that a group G is called indecomposable if
any two proper its subgroups generate a proper subgroup in G, and is
called decomposable otherwise.
If X is a class of groups, then a group G is called a minimal non-
X-group if it is not a X-group, while every proper subgroup of G is a
X-group. Every minimal non-FC-group is a minimal non-PC-group and
every torsion minimal non-PC-group is a minimal non-FC-group. It is
known that finitely generated torsion-free minimal non-PC-groups there
2010 MSC: 20F24, 20E45.
Key words and phrases: P C-group, F C-group.
2 Minimal non-PC -groups
exist (see e.g. Theorem 28.3 from [3]). V. V. Belyaev (see [4], [5] and [6])
have proved that every minimal non-FC-group with a non-trivial finite or
abelian homomorphic image is a finite extension of a divisible Černikov
p-group. F. Russo and N. Trabelsi [8] have shown that a minimal non-
PC-group with a non-trivial finite homomorphic image is an extension of
a divisible abelian group of finite rank by a cyclic group. By Corollary 2.3
of [8], a locally graded minimal non-PC-group is not finitely generated.
In this way we study the problem:“Are there non-torsion locally graded
minimal non-PC-groups?" and give the answer.
Theorem 1. Let G be a non-perfect group. Then G is a minimal non-
PC-group if and only if it is a minimal non-FC-group.
From this, in particular, it holds that every non-perfect minimal non-
PC-group has a non-trivial finite homomorphic image, and so it is torsion.
The question about the structure of perfect locally graded minimal non-
FC-groups discussed by V. V. Belyaev (see [5], [6] and [7]), M. Kuzucuoğlu
and R. E. Phillips [10], F. Leinen [11]. It is proved (see [6] and [10]) that
every perfect locally graded minimal non-FC-group must be a p-group.
In this way we prove the following
Theorem 2. A perfect locally graded minimal non-PC-group is an inde-
composable countable locally finite p-group.
Throughout this paper, p will always denote a prime, Cp∞ the quasi-
cyclic p-group, Z the integer numbers ring. For a group G, G′ will indicate
the derived subgroup, Z(G) the center, CG(H) the centralizer of H in G,
〈x〉G the normal closure of a cyclic subgroup 〈x〉 in G and Gn the subgroup
generated by the n-th powers of all elements in G.
Any unexplained terminology is standard as in [12] and [13].
2. Non-perfect minimal non-PC-groups
Lemma 1. Let G be a minimal non-PC-group. If H is a normal subgroup
of finite index in G, then G/H is a cyclic p-group for some prime p.
Proof. See [8, Lemma 3.3].
In the next we need the fact (which contains in Theorem 1.2 of [8])
that a minimal non-PC-group with a non-trivial finite homomorphic
image contains a proper divisible abelian subgroup. But the proof of
Theorem 1.2 from [8] (see its part (i)) depends on the fact that any
O. D. Artemovych 3
residually finite group, whose finite quotients are cyclic of prime-power
orders, must be finite (that is false). Therefore preliminary we prove the
following
Lemma 2. Let G be a minimal non-FC-group. If G is non-perfect, then
its derived subgroup G′ is divisible abelian and G/G′ is a cyclic p-group.
Proof. Assume, by contrary, that G′〈x〉 is proper in G for any its element x.
a) Suppose that, for every x ∈ G, G′〈x〉 is contained in some maximal
subgroup M of G. Then 〈x〉G 6 M and there exists an element b ∈ M
such that
〈x〉G = xM · 〈b〉 = 〈x〉M · 〈b〉M .
Since quotient groups M/CM (〈x〉M ) and
CM (〈x〉M )/(CM (〈x〉M )
⋂
CM (〈b〉M ))
are polycyclic-by-finite, M/(CM (〈x〉M )
⋂
CM (〈b〉M )) is also polycyclic-
by-finite. Then, in view of
CM (〈x〉M )
⋂
CM (〈b〉M ) 6 CM (〈x〉G),
we obtain that M/CM (〈x〉G) (and so G/CG(〈x〉G)) is a polycyclic-by-finite
group, a contradiction.
b) Now assume that there is an element x ∈ G such that G′〈x〉 is a
proper subgroup of G that is not contained in a maximal subgroup of G.
This means that G/G′〈x〉 is a divisible abelian group. If it is decomposable,
then G = G1G2 is a product of two proper normal PC-subgroups G1 and
G2, each of which contains the derived subgroup G′. If the quotient group
G/G′〈x〉 is indecomposable, then it is quasicyclic.
In the first case suppose that i, j ∈ {1, 2}, i 6= j and aj is a non-trivial
element of Gj . Then the quotient group
G/CGj
(〈aj〉Gj )Gi
is polycyclic-by-finite and it not contains proper subgroups of finite index.
Hence
G = CGj
(〈aj〉Gj )Gi.
Since Gi〈aj〉 is a proper PC-subgroup in G, G/GG(〈aj〉G) is polycyclic-
by-finite. But it is divisible, and therefore aj ∈ Z(G). This means that G
is abelian, a contradiction.
4 Minimal non-PC -groups
In the second case G/G′〈x〉 is a quasicyclic p-group. If the derived
subgroup G′ not contains proper subgroups of finite index, then we obtain
that a PC-subgroup G′〈x〉 is abelian for any x ∈ G, which leads to a
contradiction. Thus (G′)n is a proper subgroup in G′ for some positive
integer n, and so
A = G/(G′)n
is a torsion (and consequently locally finite) group every proper sub-
group of which is a FC-group. By results from [4], A is a FC-group, a
contradiction.
Thus the quotient group G/G′ is cyclic. By Lemma 1, G/G′ is a p-
group for some prime p and the derived subgroup G′ not contains proper
subgroups of finite index. From this it follows that G′ is a divisible abelian
group.
Lemma 3. Let G be a group with a non-trivial finite homomorphic image.
If G is a minimal non-PC-group, then it is torsion.
Proof. By Lemmas 1 and 2,
G = G′〈a〉,
where G′ is a divisible abelian group, apk
∈ G′ with some a ∈ G, a prime
p and a positive integer k.
The torsion part of G′ is normal in G. Therefore, without loss of
generality, we can assume that the derived subgroup G′ is torsion-free.
Let us t ∈ GG′(a) and n is a positive integer. Then there exists an element
x ∈ G′ such that
t = xn
and
[x, a]n = [xn, a] = [t, a] = 1.
Hence [x, a] = 1 and x ∈ GG′(a). This gives that GG′(a) is a divisible
subgroup. Since any divisible PC-group is abelian, the centralizer of
aGG′(a) in the quotient group G/GG′(a) is trivial. Therefore, without
loss of generality, we can assume that GG′(a) = 〈1〉 is trivial.
Let r and s be different primes. Since G′ is a Z[G/G′]-module, by
Lemma 2.3 of [9] it contains a submodule N such that G′/N is a torsion
group, which has some elements of orders r and s. Hence G′/N = A1 ×A2
is a group direct product of a non-trivial r-subgroup A1 and a non-trivial
O. D. Artemovych 5
r′-subgroup A2. Let B be an inverse image of A1 in G. Then H = B〈a〉
is a proper normal subgroup of G and the intersection
CH(a)
⋂
B = 〈1〉
is trivial. Inasmuch as B/N = A1 is a non-trivial divisible group and
CH(〈a〉H) 6 CH(a),
the quotient group H/CH(〈a〉H) is not polycyclic-by-finite, a contradiction.
Hence G′ is a torsion subgroup.
Proof of Theorem 1. The assertion follows from Lemmas 2 and 3.
3. Perfect minimal non-PC-groups
Lemma 4 ([1]). A group G is a PC-group if and only if, for every finite
subset ∅ 6= X ⊆ G, its normal closure 〈X〉G is a polycyclic-by-finite
group.
Lemma 5. A locally graded minimal non-PC-group G is countable.
Proof. Since G is not a PC-group, in view of Lemma 4 there is a non-
trivial element g ∈ G such that 〈g〉G is not polycyclic-by-finite. Then
〈g〉G (and consequently [G, 〈g〉]) is not finitely generated. Therefore there
exists an infinite properly ascending chain of subgroups
〈g〉 < 〈g, t1〉 < · · · < 〈g, t1, . . . , tn〉 < · · ·
such that
tn ∈ [G, 〈g〉]
and
tn /∈ 〈g, t1, . . . , tn−1〉 (n ∈ N).
Let H = 〈tn | n ∈ N〉 and a subgroup K be generated by g and those
elements in G involved in the expressions of all tn. Then
H 6 〈g〉K
6 K,
and so K is not a PC-group. Hence K = G and G is countable.
Lemma 6. A perfect locally graded minimal non-PC-group G is a locally
finite p-group.
6 Minimal non-PC -groups
Proof. a) Let H be a finitely generated subgroup of G. Then H is proper
(and consequently polycyclic-by-finite) subgroup in G. Let K be a normal
polycyclic subgroup of finite index in H. By Proposition 1.3.7 of [12], its
normal core
KG =
⋂
g∈G
g−1Kg
has a finite index in H. Then an image H of H in the quotient group
G = G/KG is contained in the center of G. Hence G is a locally solvable
group.
b) Let A/B be a chief factor of G. Without loss of generality, we can
assume that B = 〈1〉 is trivial. Then A is an elementary abelian p-group
for some prime p and 〈x〉G 6 A for any x ∈ A. Since 〈x〉G is finite, we
conclude that G = CG(〈x〉G), and so x ∈ Z(G). This means that every
chief factor is central in G, and therefore G is a locally nilpotent group.
This yields that G is a locally finite p-group.
Lemma 7. A perfect locally graded minimal non-PC-group G is inde-
composable.
Proof. Let us 1 6= g ∈ G. Since G is a countable locally finite p-group, it
is non-simple and 〈g〉G is a proper normal subgroup in G. By Lemma 4,
〈g〉G is polycyclic-by-finite.
a) If S is a proper subgroup of G and 〈S, g〉 = G, then G = 〈g〉GS is
a PC-group by Lemma 4, a contradiction. Hence 〈S, g〉 6= G.
b) Now we assume that S1, S2 are proper subgroups in G. Since
Si/(Si
⋂
CG(〈g〉G))
is a polycyclic-by-finite group (i = 1, 2), there exist t1, . . . , tn ∈ G such
that
〈S1, S2〉 6 〈CG(〈g〉G), t1, . . . , tn〉.
In view of the part a), we deduce that 〈S1, S2〉 is a proper subgroup in
the group G.
Proof of Theorem 2. The assertion holds from Lemmas 5, 6 and 7.
�
References
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conjugacy classes, Boll. Unione Mat. Ital., 7, 1990, pp.35–55.
O. D. Artemovych 7
[2] S. N. Chernikov, Groups with prescribed properties of groups, Nauka, Moscow,
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Contact information
O. D. Artemovych Institute of Mathematics
Cracow University of Technology
ul. Warszawska 24
Cracow 31-155 POLAND
E-Mail: artemo@usk.pk.edu.pl
Received by the editors: 01.02.2014
and in final form 07.02.2014.
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