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We consider the graph
$\Gamma _{\text {virt}}(G)$
whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph
$\Delta _{\text {virt}}(G)$
obtained from
$\Gamma _{\text {virt}}(G)$
by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that
$\Delta _{\operatorname {\mathrm {virt}}}(G)$
has precisely t connected components. Moreover, we study the graph
$\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$
, whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph
$\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$
obtained removing the isolated vertices is connected and has diameter at most 3.
Given a finite group G, we denote by Δ(G) the graph whose vertices are the proper subgroups of G and in which two vertices H and K are joined by an edge if and only if G = ⟨H, K⟩. We prove that if there exists a finite nilpotent group X with Δ(G) ≅ Δ(X), then G is supersoluble.
Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of $\sum _{p\in \pi (G)}d_p(G),$ where we are denoting by dp(G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.
Given a group G and a subgroup H, we let
$\mathcal {O}_G(H)$
denote the lattice of subgroups of G containing H. This article provides a classification of the subgroups H of G such that
$\mathcal {O}_{G}(H)$
is Boolean of rank at least
$3$
when G is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilisers of chains of regular partitions, and the other arises by taking stabilisers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices related to the dual Ore’s theorem and to a problem of Kenneth Brown.
For a finite group
$G$
, let
$d(G)$
denote the minimal number of elements required to generate
$G$
. In this paper, we prove sharp upper bounds on
$d(H)$
whenever
$H$
is a maximal subgroup of a finite almost simple group. In particular, we show that
$d(H)\leqslant 5$
and that
$d(H)\geqslant 4$
if and only if
$H$
occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.
For a group
$G$
, let
$\unicode[STIX]{x1D6E4}(G)$
denote the graph defined on the elements of
$G$
in such a way that two distinct vertices are connected by an edge if and only if they generate
$G$
. Let
$\unicode[STIX]{x1D6E4}^{\ast }(G)$
be the subgraph of
$\unicode[STIX]{x1D6E4}(G)$
that is induced by all the vertices of
$\unicode[STIX]{x1D6E4}(G)$
that are not isolated. We prove that if
$G$
is a 2-generated noncyclic abelian group, then
$\unicode[STIX]{x1D6E4}^{\ast }(G)$
is connected. Moreover,
$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=2$
if the torsion subgroup of
$G$
is nontrivial and
$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=\infty$
otherwise. If
$F$
is the free group of rank 2, then
$\unicode[STIX]{x1D6E4}^{\ast }(F)$
is connected and we deduce from
$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(\mathbb{Z}\times \mathbb{Z}))=\infty$
that
$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(F))=\infty$
.
Given a finite group
$G$
, the generating graph
$\unicode[STIX]{x1D6E4}(G)$
of
$G$
has as vertices the (nontrivial) elements of
$G$
and two vertices are adjacent if and only if they are distinct and generate
$G$
as group elements. In this paper we investigate properties about the degrees of the vertices of
$\unicode[STIX]{x1D6E4}(G)$
when
$G$
is an alternating group or a symmetric group of degree
$n$
. In particular, we determine the vertices of
$\unicode[STIX]{x1D6E4}(G)$
having even degree and show that
$\unicode[STIX]{x1D6E4}(G)$
is Eulerian if and only if
$n\geqslant 3$
and
$n$
and
$n-1$
are not equal to a prime number congruent to 3 modulo 4.
The generating graph
$\unicode[STIX]{x1D6E4}(H)$
of a finite group
$H$
is the graph defined on the elements of
$H$
, with an edge between two vertices if and only if they generate
$H$
. We show that if
$H$
is a sufficiently large simple group with
$\unicode[STIX]{x1D6E4}(G)\cong \unicode[STIX]{x1D6E4}(H)$
for a finite group
$G$
, then
$G\cong H$
. We also prove that the generating graph of a symmetric group determines the group.
We discuss some questions related to the generation of supersoluble groups. First we prove that the number of elements needed to generate a finite supersoluble group G with good probability can be quite a lot larger than the smallest cardinality d(G) of a generating set of G. Indeed, if G is the free prosupersoluble group of rank d ⩾ 2 and dP(G) is the minimum integer k such that the probability of generating G with k elements is positive, then dP(G) = 2d + 1. In contrast to this, if k – d(G) ⩾ 3, then the distribution of the first component in a k-tuple chosen uniformly in the set of all the k-tuples generating G is not too far from the uniform distribution.
We construct a subgroup Hd of the iterated wreath product Gd of d copies of the cyclic group of order p with the property that the derived length and the smallest cardinality of a generating set of Hd are equal to d while no proper subgroup of Hd has derived length equal to d. It turns out that the two groups Hd and Gd are the extreme cases of a more general construction that produces a chain Hd=K1<···< Kp−1=Gd of subgroups sharing a common recursive structure. For i ∈ {1,. . .,p−1}, the subgroup Ki has nilpotency class (i+1)d−1.
It is proved that if $P_G(s)$ has an Euler product expansion with all factors of the form $1-c_i/q_i^s$ where each $q_i$ is a prime power, then $G$ is soluble.
We compare the probability of generating with a given number of random elements two almost simple groups with the same socle $S$. In particular we analyse the case $S\,{=}\,{\psl}(2,p)$.
Denote by f(n) the number of subgroups of the
symmetric group Sym(n) of
degree n, and by ftrans(n)
the number of its transitive subgroups. It was conjectured by
Pyber [9] that almost all subgroups of Sym(n)
are not transitive, that is, ftrans(n)/f(n)
tends to 0 when n tends to infinity. It is still an open question
whether or not this
conjecture is true. The difficulty comes from the fact that, from many
points of view,
transitivity is not a really strong restriction on permutation groups,
and there are
too many transitive groups [9, Sections 3 and 4].
In this paper we solve the problem
in the particular case of permutation groups of prime power degree, proving
the
following result.
A structure theorem is proved for finite groups with the property that, for some integer m with m ≥ 2, every proper quotient group can be generated by m elements but the group itself cannot.
Let G be a finite group, and let IG
be the augmentation ideal of ℤG. We denote by
d(G) the minimum number of generators for the group G,
and by d(IG) the minimum
number of elements of IG needed to generate
IG as a G-module. The connection
between d(G) and d(IG)
is given by the following result due to Roggenkamp ]14&]:
formula here
where pr(G) is a non-negative integer, called the presentation
rank of G, whose
definition comes from the study of relation modules (see
[4&] for more details).
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