On residuals of finite groups

Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] 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 the above article, 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 ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \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 S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

On two sublattices of the subgroup lattice of a finite group

Let {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let {\mathcal{L}(G)} be the lattice of all subgroups of G. A chief {H/K} factor of G is {\mathfrak{F}} -central in G if {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . Let {\mathcal{L}_{c\mathfrak{F}}(G)} be the set of all subgroups A of G such that every chief factor {H/K} of G between {A_{G}} and {A^{G}} is {\mathfrak{F}} -central in G; {\mathcal{L}_{\mathfrak{F}}(G)} denotes the set of all subgroups A of G with {A^{G}/A_{G}\in\mathfrak{F}} . We prove that the set {\mathcal{L}_{c\mathfrak{F}}(G)} and, in the case when {\mathfrak{F}} is a Fitting formation, the set {\mathcal{L}_{\mathfrak{F}}(G)} are sublattices of the lattice {\mathcal{L}(G)} . We also study conditions under which the lattice {\mathcal{L}_{c\mathfrak{N}}(G)} and the lattice of all subnormal subgroup of G are modular.

A NOTE ON FINITE GROUPS GENERATED BY THEIR SUBNORMAL SUBGROUPS

Following the theory of operators created by Wielandt, we ask for what kind of formations $\mathfrak{F}$ and for what kind of subnormal subgroups $U$ and $V$ of a finite group $G$ we have that the $\mathfrak{F}$-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the $\mathfrak{F}$-residuals of the subgroups.In this paper we provide an answer whenever $U$ is quasinilpotent and $\mathfrak{F}$ is either a Fitting formation or a saturated formation closed for quasinilpotent subnormal subgroups.AMS 2000 Mathematics subject classification: Primary 20F17; 20D35

Subgroup closed Fitting classes

In (1) we showed that a subgroup closed Fitting formation is a primitive saturated formation, and in (2) we showed that a subgroup closed and metanilpotent Fitting class is a formation. Whether or not a subgroup closed Fitting class is always a formation is a question that has plagued us ever since. The purpose of this paper is to prove

Metanilpotent fitting classes

Hawkes showed in [10] that classes of metanilpotent groups which are both formations and Fitting classes are saturated and subgroup closed; more, he characterized all such classes as those local formations with a local definition consisting of saturated formations (of nilpotent groups). In [3] we showed that those “Fitting formations” which are subgroup closed are also saturated, without restriction on nilpotent length; indeed such classes are, roughly speaking, recursively definable as local formations using a local definition consisting of such classes. It is natural to ask how these hypotheses may be weakened yet still produce the same classes of groups. Already in [10] Hawkes showed that Fitting formations need be neither subgroup closed nor saturated; and in [3] we showed that a saturated Fitting formation need not be subgroup closed (though a Fitting formation of groups of nilpotent length three is saturated if and only if it is subgroup closed).