Baer–Suzuki theorem for the solvable radical of a finite group

Gordeev, Nikolai; Grunewald, Fritz; Kunyavskiı̆, Boris; Plotkin, Eugene

doi:10.1016/j.crma.2009.01.004

Cited by 22 publications

(21 citation statements)

References 22 publications

(31 reference statements)

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“…Corollary C ( [GGKP4], [GGKP5], [Gu1], [FGG], [LXZ]). A finite group G is solvable if and only if every pair of conjugate elements of G generates a solvable subgroup.…”

Section: Let Us Quote the Main Resultsmentioning

confidence: 99%

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Characterization of Solvable Groups and Solvable Radical

Grunewald

Kunyavskiı̆

Plotkin

2013

Int. J. Algebra Comput.

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“…Corollary C ( [GGKP4], [GGKP5], [Gu1], [FGG], [LXZ]). A finite group G is solvable if and only if every pair of conjugate elements of G generates a solvable subgroup.…”

Section: Let Us Quote the Main Resultsmentioning

confidence: 99%

“…Note that essentially the same argument can be applied for various ramifications of the property P as above (cf. [GPS], [GKPS], [GGKP1]- [GGKP2], [GGKP4]- [GGKP5], [Gu1], [FGG]).…”

Section: From Solvable Groups To Simple Groupsmentioning

confidence: 99%

Characterization of Solvable Groups and Solvable Radical

Grunewald

Kunyavskiı̆

Plotkin

2013

Int. J. Algebra Comput.

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“…Using deeper representation theory, better results are possible; see [2] and [5]. As conjectured by Gordeev et al [6], which has since been announced by them independently (see [7,8]) we prove that, in fact, we can take k = 4. Our result was announced in [12].…”

Section: Introductionmentioning

confidence: 58%

Untitled

2010

Proc. Amer. Math. Soc.

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We prove that there exists a constant k with the property: if C is a conjugacy class of a finite group G such that every k elements of C generate a solvable subgroup, then C generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4. We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of k = 10. By lengthening one of our arguments slightly, we obtain a value of k = 7.

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“…We should mention, in relation to Theorem 4, that a similar, though weaker, solvability criterion was recently, and independently, published in [7,8]: a finite group G is solvable if and only if every pair of conjugate elements generates a solvable subgroup. This criterion can itself be viewed as a stronger version of another solvability criterion by Thompson [14, Corollary 2]: a finite group G is solvable if and only if every pair of elements generates a solvable subgroup.…”

Section: Theorem 4 a Group G Is Solvable If And Only If The Followinmentioning

confidence: 92%

SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

Kaplan

Lévy

2010

Bull. Aust. Math. Soc.

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We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc. 12 (1911) has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p -elements, a formation P p, p , for each prime p. We show that P 2,2 (which contains the odd-order groups) is properly contained in the solvable formation.2000 Mathematics subject classification: primaray 20D10; secondary 20E32.

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Baer–Suzuki theorem for the solvable radical of a finite group

Cited by 22 publications

References 22 publications

Characterization of Solvable Groups and Solvable Radical

Characterization of Solvable Groups and Solvable Radical

Untitled

SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

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