Minimal non-\(PC\)-groups

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Minimal non-

P C

_-groups

Orest D. Artemovych

Communicated by L. A. Kurdachenko

A b s t r a c t . _{The purpose of this paper is to prove that a} non-perfect groupGis a minimal non-P C-group if and only if it is a minimal non-F C-group. It is shown that a perfect locally graded minimal non-P C-group is an indecomposable countable locally ﬁnite p-group.

1. Introduction

A groupG is called a P C-group if the quotient groupG/CG(xG) is

polycyclic-by-finite for allx ∈ G [1]. The class of P C-groups is closed with respect to subgroups, quotients and direct products of its members and containsF C-groups (that is groups with finite conjugacy classes). Recall that a groupGis called non-perfectif the derived subgroup G′ _is proper inG, and is calledperfect 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 groupGis called indecomposableif any two proper its subgroups generate a proper subgroup inG, and is calleddecomposableotherwise.

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-_{F C}-group is a minimal non-_{P C}-group and

every torsion minimal non-P C-group is a minimal non-F C-group. It is known that ﬁnitely generated torsion-free minimal non-P C-groups there

2010 MSC:20F24, 20E45.

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exist (see e.g. Theorem 28.3 from [3]). V. V. Belyaev (see [4], [5] and [6]) have proved that every minimal non-F C-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-P C-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-P C-group is not finitely generated. In this way we study the problem:“Are there non-torsion locally graded minimal non-P C-groups?" and give the answer.

Theorem 1. Let Gbe a non-perfect group. Then G is a minimal

non-P C-group if and only if it is a minimal non-F C-group.

From this, in particular, it holds that every perfect minimal

non-P C-group has a non-trivial ﬁnite homomorphic image, and so it is torsion. The question about the structure of perfect locally graded minimal

non-F C-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-F C-group must be a p-group. In this way we prove the following

Theorem 2. A perfect locally graded minimal non-P C-group is an inde-composable countable locally finite p-group.

Throughout this paper, p will always denote a prime,C_p∞ the

quasi-cyclicp-group,Zthe integer numbers ring. For a group_G,_G′ _{will indicate} the derived subgroup,Z(G) the center,CG(H) the centralizer ofH inG, hxiG_{the normal closure of a cyclic subgroup}_hxi_in_G_and_Gn_{the subgroup}

generated by the n-th powers of all elements inG.

Any unexplained terminology is standard as in [12] and [13].

2. Non-perfect minimal non-P C_-groups

Lemma 1. LetGbe a minimal non-P C-group. IfHis a normal subgroup of finite index in G, then G/H is a cyclicp-group for some prime p.

Proof. See [8, Lemma 3.3].

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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-F C-group. IfG is non-perfect, then its derived subgroup G′ _{is divisible abelian and} _G/G′ _{is a cyclic}_p_-group.

Proof. Assume, by contrary, thatG′_hxi_{is proper in}_G_{for any its element}_x_.

a) Suppose that, for everyx∈G, G′

hxi is contained in some maximal subgroupM of G. ThenhxiG ₆_M _{and there exists an element} _b_∈_M

such that

hxiG=xM· hbi=hxiM · hbiM.

Since quotient groupsM/CM(hxiM) and

CM(hxiM)/(CM(hxiM)

\

CM(hbiM))

are polycyclic-by-ﬁnite, M/(CM(hxiM)TCM(hbiM)) is also

polycyclic-by-ﬁnite. Then, in view of

CM(hxiM)

\

CM(hbiM)6CM(hxiG),

we obtain thatM/CM(hxiG) (and soG/CG(hxiG)) is a polycyclic-by-ﬁnite

group, a contradiction.

b) Now assume that there is an elementx∈G such that G′_hxi _{is a} proper subgroup ofG that is not contained in a maximal subgroup ofG. This means thatG/G′_hxi_{is a divisible abelian group. If it is decomposable,} thenG=G₁G₂ is a product of two proper normal P C-subgroupsG₁ and

G₂, each of which contains the derived subgroupG′_{. If the quotient group}

G/G′_hxi _{is indecomposable, then it is quasicyclic.}

In the ﬁrst case suppose thati, j∈ {1,2},i6=j andaj is a non-trivial

element ofGj. Then the quotient group

G/CGj(haji Gj)_G

i

is polycyclic-by-ﬁnite and it not contains proper subgroups of ﬁnite index. Hence

G=CGj(haji Gj)_G

i.

SinceGihaji is a properP C-subgroup inG,G/GG(hajiG) is

polycyclic-by-ﬁnite. But it is divisible, and thereforeaj ∈Z(G). This means that G

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In the second case G/G′

hxi is a quasicyclic p-group. If the derived subgroupG′ _{not contains proper subgroups of ﬁnite index, then we obtain} that a P C-subgroup G′_hxi _{is abelian for any} _x _∈ _G_{, which leads to a} contradiction. Thus (G′₎n _{is a proper subgroup in}_G′ _{for some positive} integern, and so

A=G/(G′ )n

is a torsion (and consequently locally ﬁnite) group every proper sub-group of which is a F C-group. By results from [4], Ais a F C-group, a contradiction.

Thus the quotient group G/G′ _{is cyclic. By Lemma 1,} _G/G′ _{is a} _p -group for some primep and the derived subgroupG′ _{not contains proper} subgroups of ﬁnite index. From this it follows thatG′ _{is a divisible abelian} group.

Lemma 3. LetGbe a group with a non-trivial finite homomorphic image. If G is a minimal non-P C-group, then it is torsion.

Proof. By Lemmas 1 and 2,

G=G′

hai,

where G′ _{is a divisible abelian group,}_apk

∈G′ _{with some}_a_∈_G_{, a prime}

p and a positive integerk.

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 ust∈GG′(a) andnis 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 G_G′(a) is a divisible

subgroup. Since any divisible P C-group is abelian, the centralizer of

aGG′(a) in the quotient group G/G_G′(a) is trivial. Therefore, without

loss of generality, we can assume that GG′(a) =h1i 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 submoduleN such thatG′

/N is a torsion group, which has some elements of ordersr ands. HenceG′

/N =A₁×A₂

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r′_-subgroup

A2. Let B be an inverse image of A1 in G. Then H=Bhai is a proper normal subgroup ofGand the intersection

CH(a)

\

B =h1i

is trivial. Inasmuch asB/N =A₁ is a non-trivial divisible group and

CH(haiH)6CH(a),

the quotient groupH/CH(haiH) is not polycyclic-by-ﬁnite, a contradiction.

HenceG′ _{is a torsion subgroup.}

Proof of Theorem 1.The assertion follows from Lemmas 2 and 3.

3. Perfect minimal non-P C_-groups

Lemma 4 ([1]). A groupGis a P C-group if and only if, for every finite subset ∅ ₆= _X _⊆ _G_{, its normal closure} _hXiG _{is a polycyclic-by-finite}

group.

Lemma 5. A locally graded minimal non-P C-group Gis countable.

Proof. Since Gis not a P C-group, in view of Lemma 4 there is a non-trivial element g ∈ G such that hgiG _{is not polycyclic-by-ﬁnite. Then} hgiG _{(and consequently [}_G,_hgi_{]) is not ﬁnitely generated. Therefore there}

exists an inﬁnite properly ascending chain of subgroups

hgi<hg, t1i<· · ·<hg, t1, . . . , tni<· · ·

such that

tn∈[G,hgi]

and

tn∈ hg, t/ 1, . . . , tn−₁i (n∈N).

Let H =htn |n ∈ Ni and a subgroup K be generated byg and those

elements inGinvolved in the expressions of alltn. Then

H 6hgiK 6K,

and soK is not aP C-group. HenceK=Gand Gis countable.

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Proof. a) LetH be a finitely generated subgroup ofG. Then H is proper (and consequently polycyclic-by-finite) subgroup in G. LetK be a normal polycyclic subgroup of finite index in H. By Proposition 1.3.7 of [12], its normal core

KG=

\

g∈G g−₁

Kg

has a ﬁnite index in H. Then an image H of H in the quotient group

G=G/KG is contained in the center ofG. HenceGis a locally solvable

group.

b) Let A/B be a chief factor of G. Without loss of generality, we can assume thatB =h1i is trivial. ThenA is an elementary abelianp-group for some prime p and hxiG ₆_A _{for any} _x_∈_A_{. Since}_hxiG _{is ﬁnite, we}

conclude that G=CG(hxiG), and so x∈Z(G). This means that every

chief factor is central inG, and thereforeG is a locally nilpotent group. This yields that G is a locally ﬁnitep-group.

Lemma 7. A perfect locally graded minimal non-P C-group G is inde-composable.

Proof. Let us 16=g∈G. Since Gis a countable locally ﬁnite p-group, it is non-simple andhgiG is a proper normal subgroup inG. By Lemma 4,

hgiG _{is polycyclic-by-ﬁnite.}

a) If S is a proper subgroup ofG andhS, gi=G, then G=hgiG_S _is

a P C-group by Lemma 4, a contradiction. Hence hS, gi 6=G. b) Now we assume that S₁, S₂ are proper subgroups inG. Since

Si/(Si

\

CG(hgiG))

is a polycyclic-by-ﬁnite group (i= 1,2), there exist t1, . . . , tn∈Gsuch

that

hS₁, S₂i6hCG(hgiG), t1, . . . , tni.

In view of the part a), we deduce thathS₁, S₂i 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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[9] B. Bruno and R. E. Phillips, A note on groups with nilpotent-by-finite proper subgroups, Archiv Math.,65, 1995, pp.369–374.

[10] M. Kuzucuoğlu and R. E. Phillips,Locally finite minimal non-F C-groups, Math. Proc. Cambridge Phil. Soc.,105, 1989, pp.417–420.

[11] F. Leinen,A reduction theorem for perfect locally finite minimal non-F C groups, Glasgow Math. J.,41, 1999, pp.81–83.

[12] J. C. Lennox and D. J. S. Robinson,The Theory of Infinite Soluble Groups, Clarendon Press, Oxford, 2004.

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C o n ta c t i n f o r m at i o n

O. D. Artemovych Institute of Mathematics

Cracow University of Technology ul. Warszawska 24

Cracow 31-155 POLAND E-Mail: [email protected]