dc.contributor.author Aivazidis, Stefanos dc.contributor.author Müller, Thomas dc.date.accessioned 2019-05-27T09:30:52Z dc.date.available 2019-05-27T09:30:52Z dc.date.issued 2019-05-28 dc.identifier.uri http://publications.mfo.de/handle/mfo/1423 dc.description.abstract A 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 ${\mathfrak{X}}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups G with trivial $\mathfrak{X}$-radical, ${\vert}G{\vert}^{\overline{\mathfrak{X}}} > {\vert}G{\vert}^\gamma$, where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When ${\mathfrak{X}}={\mathfrak{N}}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, 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 $\mathfrak{S} en_US \subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C] dc.language.iso en_US en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2019,17 dc.subject Classes of groups en_US dc.subject Formation en_US dc.subject Fitting class en_US dc.title On Residuals of Finite Groups en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2019-17 local.scientificprogram Research in Pairs 2019 en_US local.series.id OWP-2019-17 en_US local.subject.msc 20 en_US
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