Browsing by Author "Aivazidis, Stefanos"

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    Congruences Associated with Families of Nilpotent Subgroups and a Theorem of Hirsch
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Aivazidis, Stefanos; Müller, Thomas
    Our main result associates a family of congruences with each suitable system of nilpotent subgroups of a finite group. Using this result, we complete and correct the proof of a theorem of Hirsch concerning the class number of a finite group of odd order.
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    On Residuals of Finite Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Aivazidis, Stefanos; Müller, Thomas
    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 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]