For some very wide classes $\mathfrak{D}$ and
$\mathfrak{B}\subset\mathfrak{D}$ of groups, the author proves that an
arbitrary (nonabelian) group $G\in \mathfrak{D}$ (respectively $G\in
\mathfrak{B}$) satisfies the minimal condition for (nonabelian) subgroups iff
it is Cherniko

Chernikov, N. S.

English

arXiv

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

Chernikov, N.S.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Збiрник працьIн-ту математики НАН України2006, т.3, №3, 423-430UDC 519.41/47N. S.ChernikovInstitute of Mathematics of NAS of Ukraine, Kyiv, UkraineE-mail: chern@imath.kiev.uaGroups with the minimal conditionfor nonabelian subgroupsДля деяких дуже широких класiв D i B ⊂ D груп автор доводить, щодовiльна (неабелева) група G ∈ D (вiдповiдно, G ∈ B) задовольняєумову мiнiмальностi для (неабелевих) пiдгруп тодi i тiльки тодi, коливона 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 provesthat an arbitrary (nonabelian) group G ∈ D (respectively G ∈ B) satisfiesthe minimal condition for (nonabelian) subgroups iff it is Chernikov.Recall that a group G is called Shunkov, if for any its finitesubgroup K every subgroup of the quotient group NG(K)/K,generated by two its conjugated elements of prime order, isfinite. Recall that a group is called locally graded, if any itsfinitely generated subgroup 6= 1 contains a subgroup of finiteindex 6= 1 (S.N.Chernikov). The class of all periodic Shunkovgroups is wide and contains, for instance, all binary finite and2-groups. The class of all locally graded groups is very wideand contains, for instance, all abelian, locally finite, residuallyfinite groups. It is easy to see that this class is local and anygroup having a series with locally graded factors is locallyc© N. S.Chernikov, 2006424 Groups with the minimal condition . . .graded, and any residually (locally graded) group is locallygraded. At the same time, the class of all locally gradedgroups contains all groups having a series with locally finitefactors, all RN - (and so all groups of all Kurosh-Chernikovclasses), locally solvable, locally hyperabelian, radical in thesense of B.I.Plotkin, residually solvable groups.Below, as usual, π(G) is the set of all primes p for whichthe group G has a p-element 6= 1.Let A be the class of all groups G for which the followingconditions 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〉 possessesa subgroup of finite index 6= 1 or 〈g, h〉 is not periodic.(ii) If G is not periodic, then for any element g of infiniteorder of G and h ∈ G, 〈g, gh〉 possesses a subgroup offinite index 6= 1.The class A is very wide and contains, for instance, allperiodic 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 groupsand the class of all periodic A-groups is just the class of allperiodic C-groups.Let B (resp. D) be the minimal local class of groups con-taining A (resp. C) such that any group possessing a serieswith B (resp. D)-factors belongs to B (resp. D). Put B0 = A(resp. D0 = C) and for ordinals β > 0 by induction: if forsome ordinal α, β = α+1, then Bβ (resp. Dβ) be the class ofall groups which have a local system of subgroups possessinga series with Bα (resp. Dα)-factors, and if there is no such α,N. S.Chernikov 425then Bβ =⋃α<βBα (resp. Dβ =⋃α<βDα). It is easy to see thatB (resp. D) is the union of classes Bβ (resp. Dβ). It is easyto show by induction that all Bβ and, at the same time, Bare closed with respect to subgroups.The known Shunkov’s [1] and S.N.Chernikov’s [2] Theoremsestablish that a nonabelian group satisfying the minimal con-dition for nonabelian subgroups is Chernikov, if it is locallyfinite or has a series with finite factors resp. The next Theo-rem contains them.Its proof below uses some results [2] and Suchkova-ShunkovTheorem [3], which asserts that a Shunkov group with theminimal condition for abelian subgroups is Chernikov.Theorem. A (nonabelian) group G ∈ D (resp. G ∈ B)satisfies the minimal condition for (nonabelian) subgroups iffit is Chernikov.Note that Ol’shanskii’s nonabelian groups, in which allproper 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’snonabelian torsion-free groups, in which all proper subgroupsare 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 arestandard. (Remark that a group with min is periodic and anabelian 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 alsoG ∈ A. Let ζ be minimal among all ordinals α, for which Bα426 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 mayassume that G ∈ Bζ (resp. G ∈ Dζ). Suppose ζ > 0. Clearly,for some ordinal ξ, ζ = ξ+1. So G possesses a local system ofsubgroups having a series with Bξ (resp. Dξ)-factors. Everyfactor is, obviously, Chernikov or abelian (resp. Chernikov).So G is locally graded. Thus G ∈ B0 (resp. G ∈ D0), whichis a contradiction. So ζ = 0 and G ∈ A (resp. G ∈ C). In thecase of min, G ∈ C and G is, obviously, periodic nonabelianwith 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 orabelian) subgroup L such that any its proper subgroup isChernikov or abelian. We may assume that G = L. Thenfor every normal subgroup N of G, any proper subgroup ofG/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 almostsolvable. Since K 6= G, K and so H are almost abelian. IfH is almost solvable, then in view of Corollary to Theorem1 [2], Corollary 2 [2] and Lemmas 1,2 [2], it is Chernikov orabelian.(c) Show that G is periodic. Let G have elements g ofinfinite order. Since G ∈ A, every 〈g, gh〉 has a proper sub-group of finite index and so is abelian (see (b)). Then for anyu ∈ G, 〈gh : h ∈ G〉〈u〉 is non-Chernikov solvable and so isabelian (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 Gis abelian, which is a contradiction.(d) Show that G/Z(G) is Shunkov. Let K/Z(G) be a finitesubgroup of G/Z(G). In view of Kalužnin’s Theorem (see [5]),N. S.Chernikov 427CG(K/Z(G))/CG(K) is abelian. So NG(K)/CG(K) is almostabelian.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 groupNG/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)), forsome 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 quotientgroup of the group 〈g, gh〉/〈gp〉. Since G ∈ A, 〈g, gh〉 has asubgroup 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 itsatisfies min. So because of G/Z(G) is Shunkov (see (d)),by Suchkova-Shunkov Theorem [3] it is Chernikov. So Gis almost solvable, which is a contradiction (see (b)). Sosome 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, forany 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 primeorder. Suppose that this is not the case. Since A/Z(G) is pe-riodic (see (c)) and neither cyclic nor quasicyclic, it has someelements 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 thenw ∈ CG/Z(G)(b) ∩ CG/Z(G)(ch) = A/Z(G) ∩ (A/Z(G))h, whichis 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〉| < ∞. Sowith regard to (1) by Sozutov-Shunkov Theorem [6], for somenormal 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, whichis a contradiction (see (b)).The following new proposition is contained in Theorem.Proposition 1. Let G be a nonabelian group. Assumethat G is locally graded or periodic Shunkov. Then G satisfiesmin−ab iff it is Chernikov.In view of Mal’cev Theorem (see Theorem 4.2 [7]), a lineargroup over a field is locally residually finite. Further, for acommutative and associative ring R with 1 and any finitelygenerated 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 anyN. S.Chernikov 429GLn(R) is locally graded. Therefore, in virtue of Theorem,the following proposition is valid.Proposition 2. A (nonabelian) group G ⊆ AutR(M) orG ⊆ GLn(R) satisfies min (resp. min−ab) iff it is Chernikov.Finally, let E be the class of all groups G for which thefollowing 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〉 possessesa 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 followingcondition is fulfilled:If G is periodic 6= 1, then for any p ∈ π(G) and p-elementg 6= 1 of G and h ∈ CG(gp), 〈g, gh〉 possesses aD-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 imageof 〈g, gh〉 is generated by two elements. In the case when itis not abelian, by Theorem, it is finite. In the case when it isabelian, 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.430References[1] Shunkov V.P. On abstract characterizations of some linear groups // Al-gebra. Matrices and matrix groups. – Krasnoyarsk: L.V.Kirenskii In-tePhysics Sib. Dept. Acad. Sci USSR, 1970. – P. 5–54 (in Russian).[2] Chernikov S.N. Groups with the minimal condition for nonabelian sub-groups // Groups with restrictions for subgroups. – Kyiv: Naukovadumka, 1971. – P. 96–106. (in Russian).[3] Suchkova N.G., Shunkov V.P. On groups with the minimal condition forabelian subgroups // Algebra i logika. – 1986. – 25, №4. – P. 445–469 (inRussian).[4] Ol’shanskii A.Yu. Geometry of defining relations in groups. – Moscow:Nauka, 1989. – 448 p. (in Russian).[5] Kargapolov M.I., Merzljakov Ju.I. Foundations of the theory of groups. –Moscow: Nauka, 1972. – 240 p. (in Russian).[6] Sozutov A.I., Shunkov V.P. A generalization of the Frobenius theorem toinfinite groups // Mat. sb. – 1976. – 100, №4. – P. 495–506 (in Russian).[7] Wehrfritz B.A.F. Infinite linear groups. – Berlin etc.: Springer, 1973. –228 p.

Groups with the minimal condition for nonabelian subgroups

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Groups with the Minimal Conditions for Subgroups and for Nonabelian Subgroups

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)