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УДК 517.9
On a Sufficient Condition when an Infinite Group Is not Simple
Aleksei A. Shlepkin*
Institute of space and information technologies Siberian Federal University Kirenskogo, 26, Krasnoyarsk, 660074
Russia
Received 09.01.2017, received in revised form 06.04.2017, accepted 20.09.2017 We describe the conditions of existing periodic part in Shunkov group.
Keywords: Shunkov group, Groups saturated by given set of groups. DOI: 10.17516/1997-1397-2018-11-1-103-107.
V. P. Shunkov in [14] proved his famous theorem on the local finiteness and almost solvability of a periodic group G containing an involution with a finite centralizer. V. V. Belyaev in [2] on the basis of ideas from the work of V.P. Shunkov proved that any group G containing a finite involution z with a finite centralizer is locally finite. The finiteness of the involution z means that the group {z, zg) is finite for any g G G. A. I. Sozutov in [11] showed, in particular, that any group G, containing an almost perfect involution z with a finite centralizer, is not simple. An involution z of a group G is said to be almost perfect if from the condition \zzg \ = to, where g G G, implies the equality zg = zx for some involution x from G. In all the above papers, [2,11,14] it was shown that the group G is not simple (under the assumption that G is an infinite group). It was natural to consider the situation when the group G contains an involution z such that Cq(z) contains a finite number of elements of finite order, but CG(z) does not have to be finite, unlike the groups from the papers [2,11,14].
Hypothesis. Let G be an infinite group, z be an involution from G such that Cg(z) contains a finite number of elements of finite order. Then G is not a simple group.
Obviously, for the groups of Shunkov, Belyaev, and Sozutov, the above hypothesis is correct. We note that the results of V. P. Shunkov on T0-groups and groups with finitely embedded involution [10,15] are close to the formulated hypothesis. In the present paper this hypothesis was confirmed for Shunkov groups saturated with groups from the set of finite simple nonabelian groups. A group G is called a Shunkov group if for any of its finite subgroups H in the factor group Nq(H)/H any two conjugate elements of prime order generate a finite subgroup. Initially, such a group was called the conjugately birimitive finite group [9,10]. The class of Shunkov groups is very extensive and includes some mixed groups. Therefore, for each given Shunkov group G, the following question is topical: does the group G have a periodic part, e.g. do elements of finite orders in G belong to a subgroup? The nontriviality of the answer to this question is emphasized by examples of solvable Shunkov groups that do not have a periodic part (see for example [4]).
* shlyopkin@gmail.com © Siberian Federal University. All rights reserved
Theorem. Suppose that the Shunkov group G is saturated by groups from the set of finite simple nonabelian groups, and in G there is an involution z such that CG(z) contains a finite number of elements of finite order. Then G has a periodic part isomorphic to a finite simple nonabelian group. In particular, if G is an infinite group, then G is not a simple group.
1. Definitions, preliminary results
Definition 1. The group G is saturated with groups from the set of groups X, if any finite subgroup K of G is embeddable in a subgroup M of the group G, that M is isomorphic to a group in X [12].
Definition 2. Let the group G be saturated with groups from the set of groups X. Then the set X is called the saturating set for G [7].
Definition 3. Let G be a group, X be the set of groups. Recording
G G X
means that the group G is isomorphic to some group in X. Accordingly, the record
G G X
means that the group G is not isomorphic to any group in the set X.
Definition 4. Let G be a group, K be a subgroup of G, X be the set of groups. Across
Xg(K) = {H | K < H < G,H G X}
we denote the set of all subgroups H of the group G containing the subgroup K and isomorphic to groups in the set X. If 1 is the identity subgroup of G, then
XG(1) = {H I H < G,H G X}
will denote the set of all subgroups H of the group G, isomorphic to groups in the set X. If the context is clear about which group G we are talking about, then instead of XG(K) we write X(K), and accordingly XG(1) we write X(1).
Definition 5. Let G be a group. If all elements of finite orders in G are contained in a periodic subgroup of G, then it is called the periodic part of the group G and is denoted by T(G) ( [6, pp. 90,150],).
From the theorem of V.D. Mazurov ( [8]) follows
Preposition 1. For any finite set of prime numbers n, there exists only a finite set of finite simple groups (up to isomorphism) with the property that if a prime number p divides KI, where K G , then p G n [8].
Preposition 2 (Dicman's lemma). A finite invariant set of elements of finite order in any group generates a finite normal subgroup [5].
Preposition 3 (Theorem of Brauer). There exists a finite number of finite simple nonabelian groups (up to isomorphism) with a given centralizer of involution [3].
Preposition 4. The Shunkov group with an infinite number of elements of finite order has an infinite locally finite subgroup [13].
Preposition 5. Let G be a Shunkov group, a be an element of prime order in G, x an involution in G. Then (x, a) is a finite group.
Proof. It follows from the definition of the Shunkov group that (a, ax) is a finite group. It is easy to see, that x G NG((a, ax)). Consequently, (a, ax)(x) is a finite group. Since the group (x, a) coincides with the group (a, ax)(x), then (x, a) is also a finite group. □
2. Proof of the theorem
Let G be a counterexample to the statement of the theorem, and let M be the saturating set for the group G consisting of finite simple non-abelian groups. Fix an involution z from the condition of the theorem.
Lemma 1. CG(z) has a finite periodic part T(CG(z)).
Proof. Let P be the set of all elements of finite order from CG(z). By the condition of the theorem P is a finite set. Since P is an invariant set. Then by Dicman's lemma (Preposition 2) CG(z) possesses finite periodic part of T(Cq(z)). □
Lemma 2. The group G contains infinitely many elements of finite order.
Proof. Suppose the converse. By Dicman's lemma (Preposition 2), G possesses finite periodic part of T(G). A contradiction with the fact that G is a counterexample. □
Lemma 3. The group G contains an infinite locally finite subgroup.
Proof. The statement of the lemma is a consequence of Lemma 2 and Preposition 4. □
Lemma 4. The set M(1) contains groups of arbitrarily large order.
Proof. By Lemma 3, for any natural m in the group G there is a finite subgroup Km such that \Km\ > m. By the saturation condition, Km < Lm and Lm G M(1). By the arbitrariness of the choice of m, the set M(1) contains groups of arbitrarily large order. □
Lemma 5. Let Pm(i) be the set of prime divisors of the orders of groups in M(1). Then Pm(i) is an infinite set.
Proof. Suppose the converse. Then, by Proposition 1, the orders of groups in the set M(1) are bounded in the collection. A contradiction with the assertion of Lemma 4. □
By the condition of the theorem, in the group G there exists an involution z such that CG(z) has a finite periodic part T(CG(z)) (CG(z) contains a finite number of elements of finite order). By Definition 4
M((z)) = {Mz \ Mz G M(1),z G Mz}
is the set of all finite simple nonabelian subgroups of G, containing the involution z.
Lemma 6. The set M((z)) contains groups whose order is greater than any of a given natural m.
Proof. Let {ai,a2, • • • ,ak, • • •} be an infinite set of elements of groups from the set M(1) such that \ak \ = pk is a prime number and all pk are distinct (Lemma 5). By Proposition 5, the group (z, ak) is finite for any k. In view of the saturation condition
(z,ak) < Mz G M((z)).
It follows from the definition of primes pk that for any natural m there is a pk such that pk > m. Hence, m < pk < I{z,ak)I < IMzI. □
We now complete the proof of the theorem. Let {MZ^Ik =1, 2,... } be an infinite subset of the set M((z)) such that
IM^I < IM^I < ■■■ < IM(k)I < ■■■
(Lemma 6). By Proposition 3, there exists an infinite strictly increasing sequence of natural numbers
ki < k2 < ■ ■ ■ < km < ■ ■ ■
such that
ICMZkl) (z)I < ICMk2) (z)I < ■■■ < ICMi"m) (z)I < ■■■
is an infinite strictly increasing sequence of natural numbers. This contradicts the fact that for any km, ICM(km) (z)I < IT(CG(z)I (Lemma 1). The contradiction completes the proof of the theorem.
References
[1] R.Brauer, On structure of groups of finite order, Proceedings of the International Congress of Mathematicians, 1954, 209-217.
[2] V.V.Belyaev, On groups with almost regular involution, Algebra i logika, 26(1987), no. 5, 521-535 (in Russian).
[3] R.Brauer, On the structure of groups of finite order, In: proceedings of the International Congress of Mathematicians, 1954, 209-217.
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[6] M.I.Kargapolov, Yu.I.Merzlyakov, Fundamentals of Group Theory, Moscow, Nauka, 1982 (in Russian).
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[8] V.D.Mazurov, On the set of orders of elements of a finite group, Algebra i logika, 8(1994), no. 1, 81-89 (in Russian).
[9] A.N.Ostylovsky, V.P.Shunkov, On the local finiteness of a class of groups with the minimality condition, Izuch. teorii grup, Krasnoyarsk, 1975, 32-48 (in Russian).
[10] V.I.Senashov, V.P.Shunkov, Groups with finiteness conditions, Novosibirsk, Izdatel'stvo SB RAN, 2001 (in Russian).
[11] A.I.Sozutov, On groups with almost perfect involution, Tr. IMM UrB RAS, 13(2007), no. 1, 183-190 (in Russian).
[12] A.K.Shlepkin, Conjugately double-primitive finite groups containing finite unsolvable subgroups, Third Intern. Conf. In algebra, August 23-28, 1993, Sat. Tez. Krasnoyarsk, 1993 (in Russian).
[13] A.K.Shlepkin, On conjugately biprimitively finite groups with a primary minimum condition, Algebra i logika, 22(1983), 226-231 (in Russian).
[14] V.P.Shunkov, On periodic groups with almost regular involution, Algebra i logika, 4(1972), 470-49 (in Russian).
[15] V.P.Shunkov, On a class of groups with involutions (T0-groups), Mat. Raboty, 1(1998), no. 1, 139-202 (in Russian).
Об одном достаточном условии, при котором бесконечная группа не будет простой
Алексей А. Шлепкин
Институт космических и информационных технологий Сибирский федеральный университет Киренского, 26 Б/17, Красноярск, 660074
Россия
В 'работе рассмотрены условия существования периодической части группы Шункова. Ключевые слова: группа Шункова, группы насыщенные заданным множеством групп.
