36 research outputs found
On residuals of finite groups
A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} 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 \cite{DHKL}, we establish the following generalisation: if
is a subgroup-closed Fitting formation of full characteristic
which does not contain all finite groups and is the
extension-closure of , then there exists an (optimal) constant
depending only on such that, for all non-trivial finite
groups with trivial -radical, \begin{equation} \left\lvert
G^{\overline{\mathfrak{X}}}\right\rvert \,>\, \vert G\vert^\gamma,
\end{equation} where is the
-residual of . When , the class of finite nilpotent groups, it follows that
, 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 of full characteristic such that , thus providing
applications of our main result beyond the reach of \cite[Thm.~C]{DHKL}.Comment: 19 page
On the distribution of the density of maximal order elements in general linear groups
In this paper we consider the density of maximal order elements in
. Fixing any of the rank of the group, the characteristic
or the degree of the extension of the underlying field
of size , we compute the expected value of the said density and
establish that it follows a distribution law.Comment: 20 pages, substantial corrections. Accepted for publication at The
Ramanujan Journa
On the subgroup permutability degree of some finite simple groups.
PhDConsider a finite group G and subgroups H;K of G. We say that H and K permute
if HK = KH and call H a permutable subgroup if H permutes with every
subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are
permutable. We can define, for every finite group G, an arithmetic quantity that
measures the probability that two subgroups (chosen uniformly at random with
replacement) permute and we call this measure the subgroup permutability degree
of G. This measure quantifies, among others, how close a finite group is to
being quasi-Dedekind, or, equivalently, nilpotent with modular subgroup lattice.
The main body of this thesis is concerned with the behaviour of the subgroup permutability
degree of the two families of finite simple groups PSL2(2n), and Sz(q).
In both cases the subgroups of the two families of simple groups are completely
known and we shall use this fact to establish that the subgroup permutability
degree in each case vanishes asymptotically as n or q respectively tends to infinity.
The final chapter of the thesis deviates from the main line to examine groups,
called F-groups, which behave like nilpotent groups with respect to the Frattini
subgroup of quotients. Finally, we present in the Appendix joint research on the
distribution of the density of maximal order elements in general linear groups
and offer code for computations in GAP related to permutabilityChrysovergis Endowment, under the auspices of the National
Scholarships Foundation of Greec
The -series of a -group and complements of abelian subgroups
Let be a -group. We denote by the intersection of
all subgroups of having index , for . In this
paper, the newly introduced series is investigated and
a number of results concerning its behaviour are proved. As an application of
these results, we show that if an abelian subgroup of intersects each
one of the subgroups at , then has a
complement in . Conversely if an arbitrary subgroup of has a normal
complement, then
Common transversals and complements in abelian groups
Given a finite abelian group and cyclic subgroups , , of of
the same order, we find necessary and sufficient conditions for , ,
to admit a common transversal for the cosets they afford. For an arbitrary
number of cyclic subgroups we give a sufficient criterion when there exists a
common complement. Moreover, in several cases where a common transversal
exists, we provide concrete constructions.Comment: 14 page