Let
G
be a finite solvable group and
π
(
G
)
the set of primes that divide |
G
|. If, for every
g
∈
G
\
Z
(
G
)
, we have
|
π
(
C
G
(
g
)
)
|
≤
κ
, then we prove that
|
π
(
G
)
|
≤
2
·
κ
. We will ...then present some consequences of this result.
Let
G
be a finite group of even order that has no 2-rank 1. We will prove, using only elementary methods, that there is an involution
t
∈
G
such that
|
G
|
<
|
C
G
(
t
)
|
6
.
Let G be a finite soluble group, and let h(G) be the Fitting length of G. If φ is a fixed-point-free automorphism of G, that is CG(φ)={1}, we denote by W(φ) the composition length of 〈φ〉. A ...long-standing conjecture is that h(G)≤W(φ), and it is known that this bound is always true if the order of G is coprime to the order of φ. In this paper we find some bounds to h(G) in function of W(φ) without assuming that (|G|,|φ|)=1. In particular we prove the validity of the “universal” bound h(G)<7W(φ)2. This improves the exponential bound known earlier from a special case of a theorem of Dade.
We describe sharply 2-transitive groups whose point stabilizer is a nilpotent {2,3}-group without elements of order 9 and, more generally, in which the third power of each element belongs to the ...FC-center. In particular, we will prove that these groups are finite.
Let
G
be a finite and soluble group factorized by three subgroups
A
,
B
and
C
, that is
G
=
A
B
C
=
{
a
b
c
∣
a
∈
A
,
b
∈
B
,
c
∈
C
}
, with
A
B
=
B
A
,
B
C
=
C
B
and
C
A
=
A
C
. Let
h
(
G
) denote ...the Fitting length of
G
. In this paper we prove that if
AB
and
C
are nilpotent, then
h
(
G
)
≤
h
(
B
C
)
·
h
(
C
A
)
. We also prove that if
(
|
A
|
,
|
B
|
)
=
1
and
max
{
h
(
A
B
)
,
h
(
B
C
)
,
h
(
C
A
)
}
=
h
, then
h
(
G
)
≤
h
(
h
+
1
)
.
A group
G
has all of its subgroups normal-by-finite if
H
/
H
G
is finite for all subgroups
H
of
G
. The Tarski-groups provide examples of
p
-groups (
p
a “large” prime) of nonlocally finite groups in ...which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2-group with every subgroup normal-by-finite is locally finite. We also prove that if |
H
/
H
G
| 6 2 for every subgroup
H
of
G
, then
G
contains an Abelian subgroup of index at most 8.
Let
G be a residually finite group. If
G admits an automorphism of prime order
p with finite Reidemeister number, then
G is virtually nilpotent (of class bounded by a function of
p).
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