CC BY
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Серия «Математика»
2020. Т. 32. С. 101-117
Онлайн-доступ к журналу: http://mathizv.isu.ru
УДК 519.45 MSC 20F99
DOI https://doi.org/10.26516/1997-7670.2020.32.101
On Periodic Groups of Shunkov with the Chernikov Centralizers of Involutions
V. I. Senashov1'2
1 Siberian Federal University, Krasnoyarsk, Russian Federation;
2 Institute of Computational Modelling SB RAS, Krasnoyarsk, Russian Federation
Abstract. Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The author develops the direction of characterizing the well studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. In this paper, almost layer-finite groups are characterized in the class of periodic Shunkov groups. Shunkov group is a group G in which for any of its finite subgroup K in the factor group Ng(K)/K any two conjugate elements of prime order generate a finite subgroup. We study periodic Shunkov groups under the condition that a normalizer of any finite nontrivial subgroup is almost layer-finite. It is proved that if in such a group the centralizers of involutions are Chernikov ones, then the group is almost layer-finite.
Keywords: infinite group, finitness condition, Shunkov group, Chernikov group.
The group is called layer-finite if the set of its elements of any given order is finite. The class of almost layer-finite groups is wider than the class of layer-finite groups. Any Chernikov group is almost layer-finite, whereas it is easy to give examples of Chernikov groups that are not layer-finite. In this paper almost layer-finite groups are characterized in the class of periodic Shunkov groups.
1. Introduction
Theorem. Let G be a Shunkov periodic group and the centralizer of each involution in G be a Chernikov one. If the normalizer of any nontrivial
finite subgroup of the group G is almost layer-finite, then G is an almost layer-finite group.
The author has previously proved a similar theorem for groups with the minimality condition for non-almost layer-finite subgroups [15]. Recently, a number of works have also been devoted to the study of Shunkov groups [16-18].
2. Proof of the Theorem
Let G be a Shunkov periodic group that is not almost layer-finite. Additionally we assume that the centralizers of all involutions in the group G are Chernikov's and the normalizer of any nontrivial finite subgroup of the group G is almost layer-finite.
By S we denote some Sylow 2-subgroup of G, i is the central involution from S or from the intersection of the center and the complete part of S if it is infinite (if S is infinite, then it is Chernikov's by Lemma 1 from [11], by the properties of infinite Chernikov primary groups, in them the intersection of the complete part with the center is nontrivial), H is a maximal almost layer-finite subgroup of the group G containing the infinite centralizer CG(i), which is almost layer-finite by assumption. Such a maximal subgroup exists by Zorn's lemma and by theorem 1 from [12]. The centralizer CG(i) is infinite, since otherwise, by Proposition 7 of [21], the group G would be locally finite and, by theorem 1 from [12], is almost layer-finite, that contradicts our assumption about the group G.
By the theorem from [14], we can assume that H is a not strongly embedded subgroup of G. From here, by Lemma 6 from [9] it immediately follows that H has an almost regular involution. If S be finite, then we can choose this involution from S due to Lemma 9 from [10], but if S is infinite, then by Theorem 2 of [11] it contains infinitely many involutions, among which by Lemma 8 from [10] there is an almost regular in H involution. Fix for this involution notation j. We denote by R(H) a layer-finite radical of H.
Let K be a subgroup of H generated by all involutions with infinite centralizers in H.
Lemma 1. If H \ R(H) does not contain involutions conjugate with i in G, then H = CG(i) and Sylow 2-subgroups in R(H) are locally cyclic or generalized quaternion groups.
Proof. If S is a finite group, then repeating the reasoning from the beginning of the proof of the Theorem from [10] we obtain the assertion of the lemma.
Now suppose that S is infinite. By Theorem 2 of [11] S is an extension of a quasi-cyclic 2-group by a reversing automorphism. Since R is a layer-finite group, S П R is a quasi-cyclic 2-group. □
Lemma 2. At least one of statements is valid: 1) S is a 8th order dihedral group, and i, j are conjugate in G; 2) H = CG(i) and Sylow 2-subgroups from R(H) are locally cyclic or generalized quaternion groups.
Proof. If H\ R(H) does not possess involutions, conjugate with i in G, then by Lemma 1 H = CG (i) and the Sylow 2-subgroups from R(H) are locally cyclic or generalized quaternion groups. The same is true if |K| = 2.
Let K = (i) x (t). By Lemma 8 from [10], the maximal elementary Abelian subgroup R in S has an order 4, and since |CG(i)| = then t £ R (since t £ CG(j)).
Suppose that j = g-1ig and D = H n Hg. Let V be a Sylow 2-subgroup of D and R < V, P, Q are Sylow 2-subgroups from H, Hg, respectively, and V = P n Q. Obviously R < Z(V) (since i £ Z(V), so we select j £ Z(Vg) also, V, Vg are conjugate in D, hence Vg = Vh,ih = j and R is a maximal subgroup in V). Since K < P and t £ CG(j), then V = P, similarly to V = Q. Hence from the normalizer condition in nilpotent groups Ng(V) does not contained in H. Obviously R < L = NG(V).
If there was no element in L that induces an automorphism of 3-th order in R, then L = CL(R)(d), where d £ P < H and Cl(R) < CG(i) < H. Therefore, L < H, contrary to what was proved above. So in NG(V) there is an element that induces an automorphism of order 3 in R. If V had an element of order 4, then it could be chosen in V so that b2 = j, and since |K| = 4, K < H, b £ H, then b2 = j implies t £ K < CG(j) contrary to what was proved above. This contradiction means that R = V = CP(j).
Further, P is a dihedral group or a semidihedral group [2] and K < P. Therefore, P is a dihedral group of order 8. Then, in view of the conjugacy of Sylow subgroups in H, the same is valid for S. □
Remark 1. In view of the structure of a non-Chernikov almost Abelian almost layer-finite group B we assume that the number p is chosen so that it does not divide the index |B : L(B)|, where L(B) is a nilpotent radical of the group B (this index is finite, and the set n(B) is infinite by Theorem 1.1.6 from [15]). In addition to choosing the number p, we can assume that it does not belongs to the set Un(CB(K)), where K runs through all elementary Abelian subgroups of B having in B finite centralizers (Similar to the proof of Lemma 11 from [11], it is shown that the set of non-conjugate elementary Abelian subgroups of almost layer-finite group V with finite centralizers in V is finite) in the case of Chernikov group H and in the case of non-Chernikov H the number p £ n(C# (K)) for elementary Abelian subgroups K of H with finite centralizers in H.
We fix a notation. In the future we will talk about the element a from B or from H of prime order chosen according to the remark.
Consider groups of the form Ln = (a, aSn, i), where i £ Z(S), sn £ CG(i), a £ G \ H is a strictly real element with respect to the involution
i (if we consider the case of the Chernikov group H, then the element a is taken from the non-Chernikov group B and the choice of its order is unlimited; if H is a non-Chernikov group, then the element a can be chosen from subgroup conjugate to H and again to choose its order there are infinitely many variants).
Such groups as shown by A.N. Izmailov (see, for example, [19]) are finite, as soon as the groups (a, aSn) are finite, and the last groups are finite since G is a Shunkov group. Denote the set of groups Ln by N. The set N is infinite, otherwise for some sequence of the elements si, s2,..., sn,... from CG(i) aS1 = aS2 = ... = aSn = ... and hence sns-1 e CG(a),n = 1,2,... and CG(a) П CG(i) is infinite, but then, by Lemma 6 from [9], a e H contrary to the choice of the element a.
Lemma 3. The subgroups of the set N are almost all semisimple.
Proof. Suppose that Sylow 2-subgroups in G are cyclic or generalized quaternion groups. Then the Sylow 2-subgroup of Ln is, by assumption, cyclic or a generalized quaternion group for any subgroup Ln of N and according to the Brauer-Suzuki theorem [3;6] Ln = O2/(Ln) ■ CLn(i). If the element a does not belong to O2> (Ln), then the element a = aO2' (Ln) is strictly real with respect to the involution i = iO2/ (Ln). But the involution i is contained in the center of the factor group Ln/O2' (Ln). Contradiction implies the inclusion of the element a in O2/ (Ln). Obviously the same is true for the element aSn. Then, in view of generating Ln by elements a, aSn, i, it has the structure O2/ (Ln)A(i), that is, it is solvable by the Feit-Thompson theorem.
Suppose, that L1, L2,..., Ln,... is an infinite sequence of different subgroups from N, where Ln = (a,aSn,i) and Ln has a nontrivial elementary Abelian subgroup Vn, normal in Ln, n = 1,2,... We represent Vn as Vn = Zn x Fn, where Zn = CG(i) П Vn, and if |Fn| are odd, then Fn = (h e Vn | hi = h-1).
If in the set of subgroups of the form Vn, n = 1,2,..., there is only a finite set of different, it is obvious without breaking the generality of reasoning, we can assume that V = V1 = V2 = ... = Vn = ...
Consider the maximal almost layer-finite subgroup M in G containing Ng(V). By assumption, Ln = (a, aSn, i) < M. Consider two cases:
1) Cm(i) is infinite. Since M is a maximal almost layer-finite in G subgroup, then by Lemma 6 from [9] CG(i) contained in M. By Lemma 8 from [10], i is contained in a finite normal subgroup of M, and therefore, in a layer-finite radical R(M). The element aSn is a strictly real relative to i and contained in M. From here we get
iaSni = (aSn)-1, (aSn)-1iaSn e R(M).
Comparing these relations, we note: i(aSn )2 e R(M). Then, (aSn )2 e R(M) and, taking into account the oddness of the order of the elements
as", finally as" e R(M). Due to the infinity of the set {aSn}, n = 1,2,..., we get contradiction with the definition of a layer-finite group.
2) Cm(i) is finite. In this case, by Lemma 12 from [11] there is a normal in M subgroup U of finite index in M each whose element is strictly real with respect to i.
The element a is also strictly real with respect to i. From here following equality aha-1 = ia-1iih-1iiai = ia-1h-1ai = a-1ha or ha2 = a2h show that a is permuted with any element of U. So it belongs to a finite normal subgroup of the group M and belongs to its layer-finite radical R(M). Similar we show aSn e R(M), but this cannot be due to the infinity of the set {aSn } and by the definition of layer-finite radical.
Thus, both cases are impossible. Therefore not breaking the generality of reasoning, we can assume that the subgroups of the form Vn, n = 1,2,..., are different. Let Zn = 1 for any n. Suppose first that the set of primes p for p-subgroups Zn is finite.
Without breaking the generality of reasoning, we can assume that all Zn are p-groups by one prime number p. Due to the properties of almost layer-finite groups among them only a finite number of subgroups is not conjugate in CG(i). Therefore, without breaking the generality of reasoning it can be considered that Z = Z*1 = ... = Z^ = ..., where tn e CG(i). The set of {tn | n = 1,2,...} can be both finite, and infinite. However, the set {L^ | n = 1, 2,...} is always infinite due to the choice of the pair (a, i).
Let X be a maximal almost layer-finite subgroup in G containing NG(Z).
First, let the orders of the subgroups Vn, n = 1,2,..., be odd.
If Cx (i) is infinite, then by Lemma 8 from [10] the involution i belongs to a finite class of conjugate involutions in X. If there are infinitely many subgroups F^, we find various elements f1, /2,..., /n,... such, that /i-1i/n = /-Vfc. Then from equality = i/„/fc-1i = /„/-1 followed /| = /П,
but /j has odd prime orders. Contradiction.
If CX (i) is finite, then as above atn, aSntn e R(X), but there are infinitely many such different elements of the same order. The contradiction with layer-finiteness of R(X).
Then, in any case, we can assume, without breaking the generality of reasoning, that F = F*1 = ... = Fnn = ... and that means V = V*1 = ... = РП" = ... By including Ng(V) in the maximal almost layer-finite subgroup Wn of G, we get that i, ¿П" < W, n = 1,2,...
If the involution i belongs to the layer-finite radical R(W) of the group W, its centralizer in W is infinite and by Lemma 4 from [11], Lemma 6 from [9] W = H and atn belongs to H together with the element a (tn taken from H). If i £ R(W), then by Lemmas 8 from [10] and 12 from [11] in W there is an infinite Abelian normal subgroup of finite index in W, consisting of strictly real elements with respect to i. In this case as above, we obtain in the layer-finite radical R(W) infinitely many elements atn ,as"tn of the same order. A contradiction means that if the orders of subgroups of the
form Vn are odd, then Zn = 1 for almost all numbers n. But then for these n subgroups Vn(a)A(i) are Frobenius groups with the complement (i). In this case, by the properties of Frobenius groups, Vn(a) are Abelian groups.
Without breaking the generality, we can assume that i, Vn < NG((a)) = D, n = 1, 2,... According to the above, all elements of Vn are strictly real with respect to i, and by the assumption D is an almost layer-finite group. Since there are infinitely many subgroups Vn which consist of elements strictly real with respect to i, then there is an infinite subgroup of finite index in D, which is elementwise permutable with Vn, n = 1,2,... Then, subgroups Vn are contained in the layer-finite radical R(D) of the group D (Vn have finite index centralizers in D), and this cannot be due to its layer-finiteness, the infinity of the set {Vn} and finiteness of the set of prime divisors of |Vn | for all n in the aggregate.
Since Sylow 2-subgroups in G are cyclic or generalized group quaternions, then Vn cannot be a 2-group n = 1,2,.... Because, then in Ln there would be the only involution i centralizing the element a, and we chose it to be strictly real with respect to i.
Thus, we have the case: the set of prime numbers p for p-subgroups Zn is infinite. Then there are infinitely many prime divisors of orders of Vn.
All Zn will be contained in H, and moreover, almost all Zn are contained in R(H) since we chose them by different p, and n(H \ R(H)) is a finite set. Then in view of the infinite isolation of H the centralizers Cg(Z„) are contained in H. Then, almost all Fn are contained in H, and therefore Vn are contained in R(H). Then using Lemma 4 from [11] and the properties of layer-finite groups we get the inclusion a e H, which is impossible. If in this case Zn = 1 for any n, then again as above we obtain Vn < NG((a)) = D < U, where U is a maximal almost layer-finite group containing D, Vn < NG((as)) = Ds < Us, where Us is a maximal almost layer-finite group containing Ds. Since V has a normal Abelian subgroup of finite index consisting of strictly real elements with respect to i, then due to the infinity of n({Vn}) for almost all s U = Us and hence {as} < R(U) for the infinite set {as}. Contradiction. Now, it is only necessary to consider the case when Vn is a 2-group.
Recall that we assume that the lemma is false and Li, L2,..., Ln,... is an infinite sequence of subgroups from N and Ln has nontrivial elementary
Abelian subgroup Vn, normal in Ln, n — 1, 2,____ We represent Vn as
Vn = x Fn, where = CG(i) n Vn. Above it is shown that without breaking the generality of reasoning it can be considered that Z = Z^1 = Z22 = ... = Z^" = ... for some elements tn e CG(i).
Let Vn be a 2-group. We show that Vn does not contain involutions with infinite centralizers in H. Indeed, if j e Vn and CH(j) is infinite, then by Lemma 6 from [9] CgO') < H and |Ng(V„) : CgO' ) n Ng(V„)| < to implies NG(Vn) < H together with Ln, but it is impossible. In this way,
Vn contains only almost regular involutions in H, and by theorem from [8] and Lemma 6 from [9] Vn П H is a cyclic group, that is, Vn П H = (j}).
By Lemma 8 from [10], we have |Vn| < 4. If |Vn| = 2, then Vn = Zn, n = 1,2,..., and this case has already been considered above and we proved its impossibility. So |Vn | =4 and V^" = (j} x (kn}, where kn is an involution for suitable elements tn. As we have shown, the set {kn} is infinite. Hence, we get the infinity of the set {ikn}. From the structure of the group V„A(i} we conclude that the order of the element ikn may be equal to only 4 (it cannot be equal to 2 because of Lemma 8 from [10]).
If i is an almost regular involution in X, then by Lemma 12 from [11] in X there is a normal subgroup L1 of finite index on which i acts strictly real. By Lemma 8 from [10] almost all involutions tn are almost regular in X, then we can assume without breaking the generality of reasoning that they are all almost regular in X. Again, by Lemma 12 from [11], using the conjugacy of tn in X (see Lemma 10 from [11]), we find the normal subgroup L2 of finite index in X on which all tn act strictly real. The intersection L3 = L1 П L2 also normal in X and has finite index in it and centralizer of L3 contain elements itn. But in the almost layer-finite group X this situation is impossible (all elements itn have the same order 4) since this contradicts to Lemma 10 from [11] and to theorem on the power of classes of conjugate elements. If i has an infinite centralizer in X, then using Lemma 4 from [11] we get the inclusion X С H and, therefore, Ch(j) is infinite, this is contrary to the choice of j. □
Let L be a semisimple group, that is, does not have a soluble normal subgroup. Following [19] we denote by F(L) the socle of L, i.e. normal subgroup of the highest order of L, which is direct product of simple groups.
Lemma 4. Subgroups from the set N have a socle, isomorphic to PSL(2, q), where q > 3 is odd.
The statement of the lemma is proved in view of Lemma 3 in exactly the same way as the Theorem in [13].
Lemma 5. Set A = {L e N | all involutions of H П L are contained in R(H)} is finite.
Proof. Suppose that A is infinite and L1, L2,..., Ln,... is an infinite sequence of different semisimple subgroups of A. Then Hn = Ln П H is a strongly embedded subgroup in Ln for any n. (If Hn were not strongly embedded, i.e. HnnH* would contain involutions for b e Ln\Hn, then in HnHb would contain the involution k from R(H) the first, since A so defined, secondly, k is the image of an involution from R(H) in Hb, hence k e R(Hb), and therefore H = Hb and Ln < H, but it is impossible.)
Let Hn be some Sylow 2-subgroup Qn and i e Z(Qn). If S were not a dihedral group of order 8, then by Lemma 2 Qn would be a cyclic group or
generalized quaternion group. It is not difficult to see from the infinite isolation of H and from the properties of subgroups from A that the subgroup Qn is a Sylow group in Ln.
Then, repeating the reasoning from the beginning of the proof of the lemma when considering the case when Sylow 2-subgroups in G are cyclic or generalized groups of quaternions, replacing the conditions imposed on the Sylow 2-subgroups of the group G on the condition for Sylow 2-subgroups from Ln, we get the impossibility of this situation.
Consequently S is a dihedral group of order 8 and by Lemma 8 from [10] Qn is an elementary Abelian group of order 4, moreover Qn < Hn and
Qi = Q2 = ... = Qn = ....
By Suzuki theorem [24] Ln = (a, Qn) are isomorphic to SL(2,Q), over the field Q of characteristics 2, but this contradicts Lemma 4. □
Lemma 6. Every involution in a simple non-Abelian subgroup U of G is contained in a maximal elementary Abelian subgroup of order 4 of U.
Proof. Let U be a simple non-Abelian group. Then, by the Brauer-Suzuki theorem [3; 6], any of its involutions is contained in the elementary Abelian subgroup of the order not less than four, but by Lemma 8 from [10] the order of this elementary Abelian subgroup cannot be greater than four. □
Lemma 7. Orders of factor groups L/F (L),L e N, limited in aggregate.
Proof. Suppose that the lemma is false. In this case there is a sequence Li,L2, ...,Ln,... for which |Ln/F(Ln)| grows with the number n.
Let Sn be a Sylow 2-subgroup of Ln and i e Sn. By Lemma 2.15 from [19] Qn = Sn n F(Ln) is a Sylow 2-subgroup in F(Ln) and Ln = (Qn)F (Ln), n = 1,2,... Because Ln = (a, as", i), where sn e CG(i), iai = a-i, then i e F(Ln). (If i e F(Ln), then a e F(Ln) otherwise the order of the group Ln/F(Ln) did not grow. At the same time, from the normality of F(Ln) in Ln follows a-iia = a-2i e F(Ln), but a-2 e F(Ln), since a is an element of odd order and from a-2 e F(Ln) would get a e F(Ln), but it is impossible.)
By Lemma 8 from [10] the lower layer of Z(Qn) is a subgroup of order < 4. If =4, then by Lemma 6 all involutions of Qn are contained in < (Qn). And since the orders of the factor groups (Qn) grows by virtue of the isomorphism theorem together with the number n, then Sn is a dihedral group of order 8 and Qn = Rn, n > q for some number q. The subgroup (Ln)(Rn) is strongly embedded in F(Ln) and according to Suzuki theorem [24] F(Ln) is isomorphic to SL(2, Q), where Q is a field of characteristic 2, n > q. But this contradicts Lemma 4.
If |Rn| = 2, then by Lemma 4 the Sylow 2-subgroup of F(Ln) is a dihedral group of order at least eight. If the orders of these dihedral groups are not limited in aggregate, then on the properties of linear groups the
number of prime divisors in the set n(CG (k)) for some involution k is unlimited. But then CG(k) is not Chernikov and we get a contradiction with the condition of the theorem. So, the orders of the Sylow 2-subgroups of F(Ln) are bounded in aggregate. From here according to Brouwer-Feit Theorem [1], the orders F(Ln) are also limited in aggregate. But due to the semisimplicity of Ln, this implies that the index |Ln : F(Ln)| is bounded.
Thus, we obtain the boundedness of orders of factor groups of the form Ln/F(Ln), n = 1,2,... Therefore, the orders of the factor groups L/F(L), L £ N, are bounded in aggregate. □
Let, for the involution t Rt = (t) x (k) be a Klein group of orger 4 and At be a maximal almost layer-finite subgroup of the group G containing CG(t). Obviously Rt < At and t belongs to the layer-finite radical R(At) of the group At. If C_At (k) is finite, then Rt is called highlighted. If the involution t is unique in R(At), then t £ CAt(R(At)).
Lemma 8. The highlighted subgroup is a Chernikov one.
Proof. Let t be an involution, At be a maximal almost layer-finite subgroup of G containing CG(t) (recall that, by the condition of the theorem, involutions centralizers in G are Chernikov). As shown earlier, CG(t) is an infinite group, then At has a nontrivial normal in At subgroup K generated by all involutions with infinite centralizers in At. Since K is a finite group, the index |NAt (K) : CAt (K)| is finite. By the inclusion CAt (t) < CAt (K) we see that the index |NAt (K) : Ca4 (t) n NAt (K)| is finite. Then At is a Chernikov group as a finite extension of the Chernikov group (see Lemma 2.3 from [19]). □
Let L1,L2, ...,Ln,... be an infinite sequence of different subgroups from N (semisimple groups) and Ln n R(H) by Lemma 4 have almost regular involution jn of CH (i).
There is an infinite set of elements {c1,c2, ...,cn,...} in R(H) such that {M1, M2, ...Mn,...}, where Mn = Lnn, n = 1,2,... consists of various subgroups and in nMn contains the same highlighted subgroup Rt.
Lemma 9. If Rt is a highlighted subgroup of V = nMn, n = 1,2,..., then 1) the set = {CMn(x) | x £ Rt \ 1, n = 1,2,...} is finite; 2) the orders of subgroups of the set N are bounded in aggregate.
Proof. First we prove 1). Let {CMn(t) | n = 1,2,...} be infinite. Let us prove that in this case the orders of the subgroups Mn n R(At) are not bounded in aggregate.
Suppose that the orders of the subgroups R(At) n Mn are bounded in aggregate. From here and from the inclusion (t) < CAt(R(At)) obviously follows boundedness in aggregate of orders |CMn(t)|,n = 1,2,..., and since Rt < CG(t) < and Rt is a highlighted subgroup, then some involution
u of Rt induces in a normal subgroup Bt of finite index in R(At) automorphism, which translates any element from Bt to the inverse (Lemma 12 from [11]). We represent At = BtQ, where Q is a finite subgroup from At and Rt < Q. From here due to the boundedness of the orders |CMn(t)|,n = 1, 2,..., infinity of {CMn(t) | n = 1, 2,...}, layer-finiteness of R(At) and the finiteness of the index |R(At) : | implies the existence of such the number q such that CMq (t) has an element d representable as d = br, where b e Bt, r e Q, |b2|Q|| > |Cm,(t)| and bu = b-1.
Based on such a representation of the element d, we write dud-1 = b-1rur-1b-1. Obviously, rur-1 e CAt(Bt) and b-1 e R(At),r e Q,u e Rt < Q, and therefore dud-1 = b-2rur-1 e CMq (t) and rur-1 e Q n CAt (Bt). Then (dud-1)|Q| = b-2|Q| e Cm, (t), but it is impossible.
Hence |R(At) n Mn| are not limited in aggregate. Choose in R(At) Abelian normal subgroup Ct of finite index, moreover, t e Ct (such exists by Lemma 12 from [11]). It is obvious that |Ct n Mn| are also not limited in aggregate.
Since At by Lemma 8 is a Chernikov group, then Ct satisfies the minimal condition and {Mn} has an infinite subset C1, and At has such a subgroup X1, that for any subgroup L e C1 X1 coincides with the subgroup from Ct n L generated by all elements of prime order from Ct n L. For the same reasons C1 has such an infinite subset C2, and Ct such a subgroup X2 > X1, that for any subgroup L e C2 the factor group X2/X1 coincides with the subgroup generated by all elements of prime orders from (Ct n L)/X1. Thus we build a chain C1 > C2 > ... and accordingly a chain X1 < X2 < ...
Let k be an involution from Rt \ (t}. By the condition = At and t is an almost regular in A^. Then, by Lemma 12 from [11], t induces in some normal subgroup Ck of finite index from A^ an automorphism, translates all elements into inverse (we can assume that Ck is chosen from R(Ak)).
Let's assume that the set {CL(k) | L e Cn} is infinite for any n. Then the orders of the subgroups Ck n L as we showed in the case of t and Bt not limited in aggregate. Since Ak is Chernikov there is a finite subgroup in Ct of odd order W = 1, which starting with some number q is contained in a some subgroup from Cn for any n. Obviously the subgroup T = (X, Rt, W) is locally finite. In particular, we have tct = c-1 for any element c e W < Ck. Since (X, Rt} < At n T, then by Lemma 4 from [11] T < At and, hence W < At. Since any element c from W has an odd order and is strictly real with respect to t, then taking into account the normality of the layer-finite radical R(At) of the group At and the uniqueness of the involution t in R(At) we get for any element a from R(At): c-1ac = t-1ctac = t-1 cac-1t or c-2ac = a, which implies (the order of the element c is odd) that c e CG(R(At)) and it means c e R(At). Then W < R(At) n R(Ak) and by Lemma 4 from [11] we get At = Ak. The contradiction means that for an
involution k starting from some number q set {CL(k) | L e £n,n > q} is finite.
Let Ei be some subgroup from Cq, E2 be some subgroup from Cq+i etc. such that < En. By the above, {CEn(x) | x e Rt \ (t),n = 1, 2,...} is finite. It can be considered without breaking the generality of reasoning, that = Mn, i.e. B = {CMn (x) | x e Rt\(t)} is finite, moreover Rt < Mn by assumption, Xt < Mn by the construction of the chain {Mn}.
Let F(Mn) be a socle of Mn. By Lemma 12 all F(Mn) contain involutions. In addition, F(Mn) n Rt = 1. If not, and if there is a q such that F(Mq) n Rt = 1, then due to the properties of primary groups and normality of F(Mq) in Mq, the subgroup F(Mq) is nontrivial intersects with the center of a Sylow 2-subgroup that contains Rt, in particular, there is an involution z centralizing Rt. But then Rt < CG(z), r e Rt means that in CG(z) there is an elementary Abelian subgroup of order 8, one of the involutions of which is almost regular in a maximal almost layer-finite subgroup in G, containing CG(z) (Rt is chosen so), and this contradicts to Lemma 8 from [10].
Thus, F(Mn) n Rt = 1. Since F(Mn) is simple and the orders of the factor groups Mn/F(Mn) are limited in aggregate (Lemmas 4, 7), then we can assume that < F(Mn) and |F(Mn)| grow with n. If the set {F(Mra) | M e N,t £ F(M)} is infinite, then F(M) n Rt = 1 implies unboundedness in the aggregate of orders of subgroups of the set Bfc = {F(M) | M e N, t e F(M), k e F(M)}, where k is an involution from Rt \ (t). And since B is a finite set, then |CF(M)(k)| for F(M) e Bk are limited in aggregate. But then |F(M)| are also limited in aggregate by Brouwer-Fowler Theorem [2] and by Lemma 4. Contradiction. Therefore, without breaking the generality of reasoning, we can assume that t e F(Mn), and the rest involutions of Rt are not contained in F(Mn).
Let Qn be a k-invariant Sylow 2-subgroup of F(Mn) containing t (such can be found due to conjugacy of primary Sylow subgroups in Mn and Frattini lemma [4]). In the center of QraA(k) obviously there is an involution. It cannot be different from t, since this would contradict to Lemma 8 from [10]. Consequently, t e Z(Q n). At the same time, in view of Lemma 6, t contains in Pn = (t) x (vn) < Qn, where is an involution, n = 1, 2,.... If in {Pn} exists infinitely many non-highlighted subgroups, it would be possible to consider that Pn is non-highlighted for any n. Then, by Lemma 2, the Sylow 2-subgroups from Mn are a dihedral groups of order 8 and Qn = Pn, moreover, the subgroups (Mn)(Pn) are strongly embedded respectively in F(Mn). Hence by Suzuki theorem [24] we obtain a contradiction with Lemma 4. We assume that Pn is a highlighted subgroup for any n. Since
e CG(t) < At, then by Lemma 10 from [11] we will assume that v = < = ... = v^" = ..., where e Ct < CG(t). However, < Ct, which means < = Un. In view of the choice of {Mra} the orders of these
subgroups grow infinitely. Therefore {Un} consists of various subgroups and their intersection contains a highlighted subgroup P = (v} x (t}.
Repeat for P and for {Un} the same reasoning with respect to Rt and to the set N. Based on this reasoning, let us prove the existence in {Un} of an infinite subset K such that the set {C^(v) | U e K} is finite. And since v e F(U), and F(U) is a simple group by Lemma 4, then, by Brouwer-Fowler Theorem [2] and Lemma 7, the orders of the subgroups from K are bounded in aggregate. But then, given the equalities |Un| = |Mn|, n = 1,2,..., we obtain a contradiction with unboundedness in aggregate of orders of subgroups from the set {Mn}. This contradiction means that the set {CMn (x) | x e Rt \ {1}, n = 1,2,...} is finite and assertion 1 is proved.
Let us prove 2. Let the orders of subgroups from Mn be unbounded in aggregate. As elements of the set L we choose subgroups of N for whose orders the equality holds |M11 < |M2| < ... and the intersection nMn has the highlighted subgroup R = (i} x (j}.
By 1 Bj = {CMn(x) | x e Rt \ {1}, n = 1,2,...} is finite and, as shown above, FMn nR = 1, n = 1,2,... But then without breaking the generality of reasoning, we will assume that the same involution k from R belongs to all Mn. Since Bj is a finite set, then by Brouwer-Fowler Theorem [2] and Lemma 7 imply limitation of orders of subgroups from N contrary to the choice of the set L from N. Thus 2 is proved. □
Proof of the theorem. We first prove that the set N has so infinite subset L such that V = nM, M e L, is a strongly embedded subgroup in each subgroup of L. Let A1 be an arbitrary infinite subset of N, V = nB1, T1 = NBl (V1), B1 e A1, Q1 be a Sylow 2-subgroup of V containing R = (i} x (j}. By Lemma 8 from [10] the intersection R n Z(Q1) has the involution t1. Let A1 be a maximal almost layer-finite subgroup of G containing CG(t1), Y1 = A1 n B1, B1 e A1, P1 = (t1} x (z1} is a subgroup of order 4 from Q1. Since R is a highlighted subgroup by Lemma 9 the set
{CSl (x) | x e R \ 1, B e A1} (2.1)
is finite. Based on Lemma 12 from [11] and Lemma 2 it is easy to get an idea to represent the subgroup A1 in the following form A1 = Ca1 (t1)C^1 (k), where k is an involution from R \ (t1}. From here and from finiteness of the set 2.1 implies the finiteness of the set
{Y1 | B e A1}. (2.2)
If P1 is a non-highlighted subgroup, then by Lemma 2 CB1 < Y1, x e P1 \ 1. If P1 is a highlighted subgroup, then, by Lemma 9, the set {CB1 (x) | x e P1 \ 1, B1 e A1} is finite. From here and from the finiteness of the set 2.2 it follows finiteness of a set
{CB1 (x) | x e P1 \ 1, B1 e A1} (2.3)
for any subgroup of the form Pi from Qi. On the Frattini lemma Ti = NTl(Qi)Vi, and since R < Qi, then by Lemma 5 NG(Qi) is finite. Hence the set is finite:
If at least one of the subgroups belonging to finite sets 2.1-2.4 not contained in any subgroup of some infinite subset of Ai, then obviously in Ai exists such an infinite subset of A2 that V2 = nB2 = Vi, B2 e A2, Vi < V2.
Let T2 = NB2 (V2), B2 e A2, Q2 be a Sylow 2-subgroup of V2 and Qi < Q2. According to Lemma 8 from [10] the intersection of R n Z(Q2) has an involution t2. Let also A2 be a maximal almost layer-finite subgroup of G, containing Cg^), Y2 = A2 n B2, B2 e A2, P2 = (t2) x (Z2) is a subgroup of order 4 from Q2. Using the same arguments used in justifying of the finiteness of sets 2.1-2.4, we prove the finiteness of the sets
Regarding the set of A2 and subsets 2.5-2.8 reason like the previous case, etc. As a result, we get in G a strictly increasing chain of subgroups Vi < V2 < ... < Vr < ... and, accordingly, the chain Qi < Q2 < ... < Qr < ...
Since, by Lemma 9, the orders of subgroups from N are bounded in aggregate then the specified chains will terminate at the finite number r, that is, the set Ar is the last member of a strictly decreasing series Ai D A2 D ... D Ar has such an infinite subset of L, that:
1) V = nM, M e L and Nm(V) = V, M e L;
2) if Q is a Sylow 2-subgroup of V, then Ny (Q) = Nm(Q), M e L;
3) if P is a Klein subgroup of orger 4 from V, in particular, P = R, then Cy(x) = Cm(x), x e P \{1}, M e L.
Now, based on assertions 1-3, we prove that V is a strongly embedded subgroup in any subgroup of L. Let E be some subgroup from L. By assertion 1 NE(V) = V and assume that for some element g of E \ V the intersection of V n Vg has an involution z. Let Q be a Sylow 2-subgroup of Vg and z e Q. As Chernikov p-group is ZA-group and satisfies normalizer condition [7] by assertion 3 it is easy to prove the inclusion Q < V n Vg. Since by the Sylow's theorem [5] Sylow 2-subgroups are conjugate in V, then in V there exists the element h such that Qhg = Q, and, therefore, hg e Ne(Q). But on to assertion 2 hg e NE(Q) = Ny (Q) < V and g e V contrary to the assumption g e E \ V. Therefore, V is a strongly embedded subgroup in any subgroup of L and existence of the sets L is proved.
Let the set L consist of subgroups Ci, C2,..., Cn,... such that Cn = (atn,ar",i), t„,r„ e CG(i).
{Nbi(Qi),Tbi | Bi e Ai}
(2.4)
{Cb2 (x) | x e R \ 1, B2 e A2}
{Y2 | B2 e A2} {CB2 (x) | x e P2 \ 1, B2 e A2} {NB2(Q2),Tb2 | B2 e A2}
(2.5)
(2.6)
(2.7)
(2.8)
By the definition of the set N, we can assume that all a*1,..., atn,... are different. As in the group G with its strongly embedded subgroup H and some involution i from H with the condition (i, ig), g £ G \ H is finite any element g of G \ H has a representation g = hj, where h £ H, j is an involution of G \ H [23], atn = hnin, where hn £ V, in, n = 1,2,... are involutions from Cn\ V, Dn = VnVin is a group of odd order. Since V is a finite group, then we assume that h = h1 = h2 = ..., D = D1 = D2 = ....
Consider the group U = NG(D). As proven above, in £ U and the set {in | n = 1,2,...} is infinite. By the conditions of the theorem the group U is almost layer-finite. Involutions i1, i2,..., in,... in view of Lemma 8 from [10] can be considered not belonging to R(U). Further, R(U) has a finite index in U, and the set {in | n = 1,2,...} is infinite, then we can assume that all this set is selected from one adjacent class R(U)i1. Then from i1 = rnin follows i1in = rninin = rn £ R(U). This means in view of the layer-finiteness of R(U) unboundedness in aggregate of the orders of the elements i1in. Then the orders of the elements a- 1 an = i1h 1hin = i1in is also unlimited in aggregate. Hence the orders of the groups (a*1 ,atn, i) is also unbounded in aggregate contrary Lemma 9. The obtained contradiction proves the theorem. The theorem is proved.
3. Conclusion
The main result of the work involves characterizing almost layer-finite groups in the class of periodic Shunkov groups. It is proved that if in Shunkov periodic group the centralizer of every involution is a Chernikov one and the normalizer of any nontrivial finite subgroup of the group is almost layer-finite, then the group is an almost layer-finite group.
References
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Vladimir Senashov, Doctor of Sciences (Physics and Mathematics), Professor, Siberian Federal University, 79, Svobodniy av., Krasnoyarsk, 660041, Russian Federation; Institute of Computational Modelling SB RAS, 50/44, Akademgorodok, Krasnoyarsk, 660036, Russian Federation, email: [email protected], ORCID iD: 0000-0001-8487-5688.
Received 14.01.2020
О периодических группах Шункова с черниковским централизатором инволюции
В. И. Сенашов1,2
1 Сибирский федеральный университет, Красноярск, Российская Федерация;
2 Институт вычислительного моделирования СО РАН, Красноярск, Российская Федерация
Аннотация. Слойно конечные группы впервые появились без названия в статье С.Н.Черникова (1945). Почти слойно конечные группы являются расширениями слойно конечных групп при помощи конечных групп. Класс почти слойно конечных групп шире, чем класс слойно конечных групп, он включает в себя все группы Черникова, в то время как легко привести примеры групп Черникова, которые не являются слойно конечно. Автор развивает направление характеризации известных хорошо изученных классов групп в других классах групп с некоторыми дополнительными (довольно слабыми) условиями конечности. В данной работе почти слойно конечные группы получают характеризацию в классе периодических групп Шунко-ва. Группа Шункова — это группа О, в которой для любой ее конечной подгруппы К в фактор-группе Ыа(К)/К любые два сопряженных элемента простого порядка порождают конечную подгруппу. Мы изучаем периодические группы Шункова с условием: нормализатор любой конечной неединичной подгруппы почти слойно-конечен. Доказано, что если в такой группе централизаторами инволюций являются черниковскими, то группа почти слойно конечна.
Ключевые слова: бесконечная группа, условие конечности, группа Шункова, группа Черникова.
Список литературы
1. Brauer R., Feit W. Analogue of Jordan's theorem in characteristic p // Ann. Math. 1966. Vol. 84. P. 119-131.
2. Gorenstein D. Finite groups. N. Y. : Harper and Row, 1968. 256 p.
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4. Каргаполов М. И. Локально конечные группы со специальными силовскими р-подгруппами : автореф. дис. ... канд. физ.-мат. наук. Молотов, 1955. 20 с.
5. Каргаполов М. И., Мерзляков Ю. И. Основы теории групп. 3-е изд., перераб. и доп. М. : Наука, 1982. 288 c.
6. Kegel O. H., Wehrfritz B. A. F. Locally Finite Groups. Amsterdam ; London : North-Holland Publ., Co., 1973. 316 p.
7. Курош А. Г. Теория групп. 3-е изд. М. : Наука, 1967. 648 c.
8. Павлюк И. И., Шафиро А. А., Шунков В. П. О локально конечных группах с условием примарной минимальности для подгрупп // Алгебра и логика. 1974. Т. 13, № 3. С. 324-336.
9. Сенашов В. И. Достаточные условия почти слойной конечности группы // Украинский математический журнал. 1999. Т. 51, N 4. C. 472-485.
10. Сенашов В. И. О силовских подгруппах периодических групп Шункова // Украинский математический журнал. 2005. Т. 57, № 11. С. 1584-1556.
11. Сенашов В. И. Строение бесконечной силовской подгруппы в некоторых периодических группах Шункова // Дискретная математика. 2002. Т. 14, № 4. С. 133-152. DOI: https://doi.org/10.4213/dm268
12. Сенашов В. И., Шунков В. П. Почти слойная конечность периодической части группы без инволюций // Дискретная математика. 2003. Т. 15, № 3. С. 91-104. DOI: https://doi.org/10.4213/dm208
13. Сенашов В. И. Характеризации групп Шункова // Украинский математический журнал. 2008. Т. 60, № 8. С. 1110-1118.
14. Сенашов В. И. О группах с сильно вложенной подгруппой, имеющей почти слойно конечную периодическую часть // Украинский математический журнал. 2012. Т. 64, № 3. С. 384-391.
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16. Шлепкин А. А. Об одном достаточном условии существования периодической части в группе Шункова // Известия Иркутского государственного университета. Серия Математика. 2017. Т. 22. С. 90-105. https://doi.org/10.26516/1997-7670.2017.22.90
17. Шлепкин А. А. О периодической части группы Шункова, насыщенной сплетенными группами // Труды ИММ УрО РАН. 2018. № 3(24). С. 281-285. https://doi.org/10.21538/0134-4889-2018-24-3-281-285
18. Шлепкин А. А. О периодической части группы Шункова, насыщенной линейными группами степени 2 над конечными полями четной характеристики // Чебышевский сборник. 2019. Т. 20, № 4. C. 399-407. https://doi.org/10.22405/2226-8383-2019-20-4-399-407
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22. Шунков В. П. О периодических группах с некоторыми условиями конечности // Доклады АН СССР. 1970. Т. 195, № 6. С. 1290-1293.
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24. Suzuki М. Finite groups of even order in which Sylow 2-groups are independent // Ann. Math. 1964. Vol. 80. P. 58-77.
Владимир Иванович Сенашов, доктор физико-математических наук, профессор, Сибирский федеральный университет, Российская Федерация, 660041, Красноярск, пр. Свободный, 79; Институт вычислительного моделирования СО РАН, Российская Федерация, 660036, Красноярск, Академгородок, 50/44, тел.: +79607626610, email: [email protected], ORCID iD: 0000-0001-8487-5688.
Поступила в 'редакцию 14-01.2020
