N-barely transitive permutation groups Alkış, Oğuz; Arıkan, Ahmet; Arıkan, Aynur
Ricerche di matematica,
06/2023, Letnik:
72, Številka:
1
Journal Article
Recenzirano
A group
G
is an
N-barely transitive group
(
NBT-group
) if
G
acts on an infinite set transitively and faithfully and all proper normal subgroups of
G
have finite orbits. We investigate the main ...properties and structure of
NBT
-groups. We give some examples in non-perfect and perfect case. Also we show that there does not exist a locally soluble perfect cofinitary
NBT
-group.
For positive integer
k
and nonabelian simple group
S
, let
S
k
be the direct product of
k
copies of
S
. We conjecture that all finite groups
G
with
cd
(
G
)
=
cd
(
S
k
)
are quasi perfect groups ...(that is;
G
′
=
G
′
′
) and hence nonsolvable groups, where
cd
(
G
)
is the set of irreducible character degrees of
G
. In this paper, we prove this conjecture for
S
∈
{
PSL
2
(
p
f
)
,
PSL
2
(
2
f
)
,
Sz
(
q
)
}
, where
p
>
2
is an odd prime number such that
p
f
>
5
and
p
f
±
1
∤
2
k
, and
q
=
2
2
n
+
1
⩾
8
.
For a finite group
G
, let the character degree set of
G
, denoted by
cd
(
G
)
, be the set of the degrees of all irreducible complex representations of
G
. In the present paper, the structure of ...finite groups whose character degree sets coincide with the character degree sets of direct product of nonabelian simple groups will be studied. For a finite group
S
and a positive integer
n
, let
S
n
be the direct product of
n
copies of
S
. We prove that if
G
is a finite group with
cd
(
G
)
=
cd
(
H
)
, where
H
∈
{
PSL
3
(
q
)
n
(
q
=
2
α
⩾
4
,
3
∤
q
-
1
)
,
PSU
3
(
q
)
n
(
q
=
2
α
⩾
4
,
3
∤
q
+
1
)
,
M
11
n
(
1
⩽
n
⩽
2
)
,
M
23
n
,
J
1
n
,
J
2
n
,
J
3
n
,
J
4
n
}
,
then
G
is a quasi perfect group. This extends the first step of Huppert’s Conjecture to the direct product of simple groups. This conjecture states that the nonabelian simple groups are uniquely determined up to an abelian direct factor by the set of character degrees.
On decomposability of finite groups Chen, Ruifang; Zhao, Xianhe
Czechoslovak mathematical journal,
09/2017, Letnik:
67, Številka:
3
Journal Article
Recenzirano
Odprti dostop
Let
G
be a finite group. A normal subgroup
N
of
G
is a union of several
G
-conjugacy classes, and it is called
n
-decomposable in
G
if it is a union of
n
distinct
G
-conjugacy classes. In this paper, ...we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.
In this note we consider certain Fitting
p-groups in which every proper subgroup satisfies an outer commutator identity and obtained some conditions for such groups to be imperfect. We also give an ...application of the main theorem to obtain an idea of the abundance of the groups under consideration.
The general solution of the functional equation
f
(
xy
) =
f
(
x
)
h
(
y
) +
f
(
y
) on abelian groups is well-known. We present methods for solving this equation on various non-abelian groups. In ...particular we treat the equation on semidirect products, then extend this treatment to solvable groups. We also find the solution on perfect groups. These results also apply to the more general equation
f
(
xy
) =
g
(
x
)
h
(
y
) +
k
(
y
).
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