On Groups with Frobenius Elements

Popov, A. A.

doi:10.1007/s10440-004-5627-z

Cited by 2 publications

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“…For example, the following proposition ensues from Theorem 3.1 in [12] and Theorem 5.1 in [2]: Proposition 2. Suppose that n, k, and q are naturals, q is a prime, (q, k) = 1, π(n) ⊆ π(k), the number n is odd, and n ≥ 665.…”

Section: Definitions and Prerequisitesmentioning

confidence: 96%

“…A nonunit proper subgroup H in a group G is isolated in G if H ∩ H g = 1 for every g ∈ G \ H; in this case (G, H) is a Frobenius pair. Also, observe the following property (see the equivalent Definition 2.1 in [2]): Proposition 1. A group G = F H is a Frobenius group with kernel F and complement H when the subgroup H is isolated in G and each nonunit element in G lies either in F or exactly in one subgroup conjugate with H.…”

Section: Definitions and Prerequisitesmentioning

confidence: 97%

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On groups with Frobenius elements

Popov

Sozutov

2015

Sib Math J

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We find the conditions under which the set-theoretic union of the kernels of two-generated Frobenius subgroups of a group G with fixed cyclic complement of order 3n is a normal subgroup in G.The article deals with groups with an H-Frobenius element a of order divisible by 3.An element a of a group G is called H-Frobenius if H is a proper subgroup of G and all subgroups of the form L g = a, a g ,Frobenius groups with complements containing a. If, moreover, the complements of all groups L g coincide with a then a is called cyclically H-Frobenius; and if H = N G ( a ) then a is called Frobenius [1, 2]. These systems of Frobenius subgroups naturally arise in the study of groups with various conditions of finiteness, factorizability, exact 2-transitivity, etc. Finite groups with Frobenius element of order > 2 were studied by Fischer [3, 4] and Aschbacher [5]. Arbitrary groups G with a cyclically H-Frobenius element a of simple odd order were considered by Shunkov [6], and in the case of |a| > 2, by Sozutov [7]. It was proved in these articles that the normal closures a G and a H of a cyclically H-Frobenius element a are Frobenius groups with complement a . Let G be a group with an H-Frobenius element a of order > 2. The main task of this article was formulated in Question 10.61 in [8]: Is the union of all kernels of Frobenius subgroups with the complement a a subgroup of G?The affirmative answer to this question was known for |a| = 2n [9] and also in the case when |a| / ∈ {3, 5} and a is finite in G [10]. Here, in the first case, the proof substantially used the abelianity of the kernels of the Frobenius subgroups L g and the uniqueness of an involution in the complement; and in the second case, the finiteness of all subgroups L g (g ∈ G).As is known, the kernel of a finite Frobenius group is nilpotent, and the subgroup in the complement generated by all elements of prime orders is either cyclic or isomorphic to the direct product of a Hall cyclic subgroup and one of the groups SL 2 (3) and SL 2 (5) (this explains why the case |a| ∈ {3, 5} is exceptional). Every subgroup of a finite Frobenius group generated by a pair of conjugate elements is again Frobenius. These properties need not be fulfilled in an infinite Frobenius group. As was proved in [11], each group is isomorphically embeddable in the kernel of a suitable Frobenius group with cyclic complement, and by [12,13], a periodic Frobenius group with abelian kernel need not be locally finite. Therefore, at present, the complete solution of Question 10.61 of [8] seems problematic to the authors. But, for the cases when the order of a divides by 3 or 5, the prognosis is more favorable due to evident progress in the study of Frobenius groups and complements generated by elements of small orders (see, for example, [14][15][16][17][18][19][20][21]). For instance, Zhurtov, Mazurov, and Churkin found a number of conditions promising for the above problem, under which a Frobenius group generated by elements of small orders is finite. The present article relies upon these ...

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Section: Definitions and Prerequisitesmentioning

confidence: 96%

Section: Definitions and Prerequisitesmentioning

confidence: 97%

On groups with Frobenius elements

Popov

Sozutov

2015

Sib Math J

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“…1. Lemma 7 and the uniqueness of an involution in the complement of a Frobenius group [7,8] imply that the specified involutions belong to J.…”

Section: Proofs Of Theorems 2 Andmentioning

confidence: 97%

“…It is obvious that M = s ik , s ik c and L = s ik , s ik d are finite Frobenius groups with cyclic complement s ik that lie in the appropriate subgroups of X i . The properties of finite Frobenius groups (for instance, see Lemma 2.1 in [8]) imply that…”

Section: Lemma 3 Implies That J Is a Unique Involution Inmentioning

confidence: 98%

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The F-local subgroups of the groups with a generalized finite element of order 2 or 4

Sozutov

Yanchenko

2007

Sib Math J

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We prove the existence of infinite subgroups with nontrivial locally finite radicals and of infinite locally finite subgroups in the groups with almost finite almost solvable elements of order 2 and 4 and in the groups with almost finite elements. This article is a continuation of [1]. Recall that by an f -local subgroup of an infinite group G we mean every infinite subgroup with a nontrivial locally finite radical [2]. We say that a mixed group G possesses a periodic part if all finite order elements of G constitute a subgroup [3]. A nontrivial finite order element a of some infinite group G is called generalized finite if for almost all elements b g of some conjugacy class b G , where b = 1, the subgroups a, b g are finite; i.e., the (a, b)-finiteness condition [3] is fulfilled. If in addition b ∈ a G then the element a is called almost finite. The results of [1] and this article yield sufficient conditions for the existence of f -local and infinite locally finite subgroups in the groups with almost finite and generalized finite elements (Questions 1.24 by Kargapolov and 2.75 by Strunkov [4]). If the (a, b)-finiteness condition is fulfilled in some group then it holds for some pair of prime order elements. As Shunkov showed [5], the case |a| = |b| = 2 and b ∈ a G is special: in this case any analog of Theorem 1 fails; see Example 1 and Theorem 1 in [1]. However, if b / ∈ a G then both involutions a and b belong to f -local subgroups that contain infinitely many finite order elements [6]. Therefore, the cases of an order 4 almost finite element and a pair (a, b) with |a| · |b| = 8 emerge as subjects of independent investigation, which is presented in this article.Theorem 1. Take some infinite group G, some order 4 element a in G, and suppose that for almost all elements a g ∈ a G the subgroups a, a g are finite and solvable. Then either G possesses finite periodic part or a belongs to some f -local subgroup that contains infinitely many finite order elements.Theorem 2. Take some infinite group G and some elements a, b in G, one of which is an involution and the other has order 4, and suppose that for almost all elements b g ∈ b G the subgroups a, b g are finite and solvable. Then either G possesses finite periodic part or at least one of a and b belongs to some f -local subgroup that contains infinitely many finite order elements.Theorem 3. Given some group G, if all finite subgroups are solvable and every quotient by a finite subgroup either is a torsion-free group or contains a generalized finite element of order > 2 then G either possesses finite periodic part or includes an infinite locally finite subgroup.

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On Groups with Frobenius Elements

Abstract: It is proved that group G contains an Abelian normal periodic complement to C G (a 2 ) if a is an H -Frobenius element a of order 4 of G. (2000): 20E25. Mathematics Subject Classification

Cited by 2 publications

References 5 publications

On groups with Frobenius elements

On groups with Frobenius elements

The F-local subgroups of the groups with a generalized finite element of order 2 or 4

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