On Residuals of Finite Groups

dc.bibliographicCitation.seriesTitleOberwolfach Preprints (OWP)
dc.bibliographicCitation.volume17
dc.contributor.authorAivazidis, Stefanos
dc.contributor.authorMüller, Thomas
dc.date.accessioned2024-10-16T16:43:27Z
dc.date.available2024-10-16T16:43:27Z
dc.date.issued2019
dc.description.abstractA theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if X is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X¯¯¯¯ is the extension-closure of X, then there exists an (optimal) constant γ depending only on X such that, for all non-trivial finite groups G with trivial X-radical, |G|X¯¯¯¯>|G|γ, where GX¯¯¯¯ is the X¯¯¯¯-residual of G. When X=N, the class of finite nilpotent groups, it follows that X¯¯¯¯=S, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that S⊂X¯¯¯¯⊂E, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C]
dc.description.versionpublishedVersion
dc.identifier.urihttps://oa.tib.eu/renate/handle/123456789/16908
dc.identifier.urihttps://doi.org/10.34657/15930
dc.language.isoeng
dc.publisherOberwolfach : Mathematisches Forschungsinstitut Oberwolfach
dc.relation.doihttps://doi.org/10.14760/OWP-2019-17
dc.relation.issn1864-7596
dc.rights.licenseDieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.
dc.rights.licenseThis document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
dc.subject.ddc510
dc.subject.otherClasses of groups
dc.subject.otherFormation
dc.subject.otherFitting class
dc.titleOn Residuals of Finite Groups
dc.typeReport
dc.typeText
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