Finite groups in which every two elements generate a soluble subgroup

Flavell, Paul

doi:10.1007/bf01884299

Cited by 32 publications

(7 citation statements)

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“…solvable), so that the pseudovarieties G nil and G sol are both defined by a set of 2-variable pseudo-identities. In the nilpotent case, this is a result of Neumann and Taylor [8] and in the solvable case, it was proved by Thompson [11], see also Flavell [4]. In fact, it is known that there exists an element u nil (a, b) (resp.…”

Section: Corollary 27 If V Is Extension-closed and ϕ Is Defined On mentioning

confidence: 80%

A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups

Coulbois¹,

Sapir²,

Weil³

2003

Publicacions Matemàtiques

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Let V be a pseudovariety of finite groups such that free groups are residually V, and let ϕ : F (A) → F (B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extensionφ of ϕ between the pro-V completions of F (A) and F (B) is also injective. In particular, if V is extension-closed, this is the case if and only if ϕ(F (A)) and its pro-V closure in F (B) have the same rank. We examine a number of situations where the injectivity ofφ can be asserted, or at least decided, and we draw a few corollaries.In this paper, we are interested in the pro-V topologies on finitely generated free groups, where V is a pseudovariety of groups (a class of finite groups closed under taking subgroups, quotients and finite direct products). These topologies were introduced in the 1950s by Hall. When V is the class of all finite groups, the finite index subgroups are exactly the open subgroups, and Hall proved [6] that every finitely generated subgroup is closed. More recent papers (Ribes and Zalesskiȋ [9], Margolis, Sapir and Weil [7], Weil [12]) focused on the problem of effectively computing the pro-V closure Cl V (H) of a given finitely generated subgroup H of a free group. It is known for instance that if V is extension-closed, then Cl V (H) has finite rank, at most equal to the rank of H [9]. In general, a finite rank subgroup may have an infinite rank closure (e.g. if V is the pseudovariety of finite abelian groups), or it may be the case that if H is a finite rank subgroup, then its closure 2000 Mathematics Subject Classification. 20E18, 20E05. Key words. Profinite topology, rank of subgroups of free groups, monomorphisms between free groups. This paper was prepared while the second author was an invited Professor at Université Bordeaux-1. The third author also acknowledges partial support from 1999 INTAS grant number 1224 Combinatorial and geometric theory of groups and semigroups and its applications to computer science.

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Section: Corollary 27 If V Is Extension-closed and ϕ Is Defined On mentioning

confidence: 80%

A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups

Coulbois¹,

Sapir²,

Weil³

2003

Publicacions Matemàtiques

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“…Since the pseudoidentity u ω = 1 only involves two variables, the validity of this conjecture would entail a result due to Thompson [38] stating that a finite group is solvable if and only if all its 2-generated subgroups are solvable. While Thompson derived this result as a corollary of his complete classification of simple groups whose proper subgroups are solvable, a proof of which extends over 410 published pages, Flavell [18] obtained a direct short and elementary proof of the same corollary.…”

Section: Tameness Of Pseudovarieties Of Groups 397mentioning

confidence: 90%

Dynamics of implicit operations and tameness of pseudovarieties of groups

Almeida

2001

Trans. Amer. Math. Soc.

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Abstract. This work gives a new approach to the construction of implicit operations. By considering "higher-dimensional" spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesskiȋ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of p-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups V, if it can be decided whether a finitely generated subgroup of the free group with the pro-V topology is dense, then V is tame.

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“…A direct proof not using the classification of minimal simple groups has been obtained by Flavell in [11]. In this section, we shall get an extension of Thompson's result above under a weaker condition, that is, we have the following theorem.…”

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confidence: 86%

Minimal inner-Σ-Ω-groups and their applications

2011

Sci. China Math.

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We introduce and study the minimal inner-Σ-Ω-groups and the minimal outer-Σ--groups. Then we give some applications and obtain some interesting results, including characterizations of nilpotent, supersolvable, solvable, and p-closed groups in terms of the join of two conjugate cyclic subgroups having the same property. Keywordsnilpotent subgroup, supersolvable subgroup, a pair of conjugate subgroups, minimal inner-Σ-Ω-group MSC(2000): 20D10, 20D15Citation: Li X H, Xu Y. Minimal inner-Σ-Ω-groups and their applications.

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Finite groups in which every two elements generate a soluble subgroup

Cited by 32 publications

References 6 publications

A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups

A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups

Dynamics of implicit operations and tameness of pseudovarieties of groups

Minimal inner-Σ-Ω-groups and their applications

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