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

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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

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

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