Affiliation:
1. Department of Mathematics , University of Brasilia , 70910-900 Brasília DF , Brazil
2. Department of Mathematics , Federal University of Goias , 75704-020 Catalão GO , Brazil
Abstract
Abstract
Let q be a prime, n a positive integer and A an elementary abelian group of order
q
r
{q^{r}}
with
r
≥
2
{r\geq 2}
acting on a finite
q
′
{q^{\prime}}
-group G.
We show that if all elements in
γ
r
-
1
(
C
G
(
a
)
)
{\gamma_{r-1}(C_{G}(a))}
are n-Engel in G for any
a
∈
A
#
{a\in A^{\#}}
, then
γ
r
-
1
(
G
)
{\gamma_{r-1}(G)}
is k-Engel for some
{
n
,
q
,
r
}
{\{n,q,r\}}
-bounded number k, and if, for some integer d such that
2
d
≤
r
-
1
{2^{d}\leq r-1}
, all elements in the dth derived group of
C
G
(
a
)
{C_{G}(a)}
are n-Engel in G for any
a
∈
A
#
{a\in A^{\#}}
, then the dth derived group
G
(
d
)
{G^{(d)}}
is k-Engel for some
{
n
,
q
,
r
}
{\{n,q,r\}}
-bounded number k.
Assuming
r
≥
3
{r\geq 3}
, we prove that if all elements in
γ
r
-
2
(
C
G
(
a
)
)
{\gamma_{r-2}(C_{G}(a))}
are n-Engel in
C
G
(
a
)
{C_{G}(a)}
for any
a
∈
A
#
{a\in A^{\#}}
, then
γ
r
-
2
(
G
)
{\gamma_{r-2}(G)}
is k-Engel for some
{
n
,
q
,
r
}
{\{n,q,r\}}
-bounded number k, and if, for some integer d such that
2
d
≤
r
-
2
{2^{d}\leq r-2}
, all elements in the dth derived group of
C
G
(
a
)
{C_{G}(a)}
are n-Engel in
C
G
(
a
)
{C_{G}(a)}
for any
a
∈
A
#
,
{a\in A^{\#},}
then the dth derived group
G
(
d
)
{G^{(d)}}
is k-Engel for some
{
n
,
q
,
r
}
{\{n,q,r\}}
-bounded number k.
Analogous (non-quantitative) results for profinite groups are also obtained.
Subject
Algebra and Number Theory
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