Characterization of Solvable Groups and Solvable Radical

Grunewald, Fritz; Kunyavskiı̆, Boris; Plotkin, Eugene

doi:10.1142/s0218196713300016

Cited by 9 publications

(11 citation statements)

References 109 publications

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“…When we specialize our results to suitable factorizations, as mentioned above, we derive descriptions of the elements in F k (G), the radical of a group G for the class N k of soluble groups with nilpotent length at most k ≥ 1, as well as the elements in the hypercenter of G modulo F k−1 (G), in the spirit of the characterization of the soluble radical in Theorem 3 (see Corollaries 3 and 4). In particular, this first result contributes an answer to a problem that is posed by F. Grunewald, B. Kunyavskiȋ and E. Plotkin in [14]. These authors present a version of Theorem 3 for general classes of groups with good hereditary properties ([14] (Theorem 5.12)), by means of the following concepts: Definition 2 (Grunewald, Kunyavskiȋ, Plotkin [14] (Definition 5.10)).…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

“…In particular, this first result contributes an answer to a problem that is posed by F. Grunewald, B. Kunyavskiȋ and E. Plotkin in [14]. These authors present a version of Theorem 3 for general classes of groups with good hereditary properties ([14] (Theorem 5.12)), by means of the following concepts: Definition 2 (Grunewald, Kunyavskiȋ, Plotkin [14] (Definition 5.10)). For a class X of groups and a group G, an element g ∈ G is called locally X -radical if g x belongs to X for every x ∈ G; and the element g ∈ G is called globally X -radical if g G belongs to X .…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

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Products of Finite Connected Subgroups

et al. 2020

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For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all a∈A and b∈B. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

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Section: Introduction and Main Resultsmentioning

confidence: 95%

Section: Introduction and Main Resultsmentioning

confidence: 95%

Products of Finite Connected Subgroups

et al. 2020

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“…There are various characterizations of finite solvable groups (some of them are discussed in [52]), among which there are those using identities. For instance, a finite group G satisfying a "short" identity of some special form must be solvable [23] (we thank the second referee for providing this reference).…”

Section: Theorem 21mentioning

confidence: 99%

“…This problem has a long history and admits many counterparts and generalizations; the interested reader is referred to [52] for a survey.…”

Section: Two-variable Sequencesmentioning

confidence: 99%

“…Note that even the toy problem of characterizing the solvable radical of a finite-dimensional Lie algebra with the help of Engellike sequences remains open in the case where the ground field is of positive characteristic; see [7,52]. We hope that algebraic-geometric machinery in the spirit of Section 3 may turn out to be useful for achieving this goal.…”

Section: Problem 73mentioning

confidence: 99%

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Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

2013

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We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

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Thompson-like characterization of solubility for products of finite groups

Hauck

Kazarin

Martínez-Pastor

et al. 2020

Annali di Matematica

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A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups G = AB with subgroups A, B such that a, b is soluble for all a ∈ A and b ∈ B. In this case, the group G is said to be an S-connected product of the subgroups A and B for the class S of all finite soluble groups. Our main theorem states that G = AB is S-connected if and only if [A, B] is soluble. In the course of the proof we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.

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Characterization of Solvable Groups and Solvable Radical

Cited by 9 publications

References 109 publications

Products of Finite Connected Subgroups

Products of Finite Connected Subgroups

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

Thompson-like characterization of solubility for products of finite groups

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