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On (???? + ½)-Engel groups
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Abstract
Let n be a positive integer.
We say that a group G is an
(
n
+
1
2
)
{(n+\frac{1}{2})}
-Engel group if it satisfies the law
[
x
,
y
n
,
x
]
=
1
{[x,{}_{n}y,x]=1}
.
The variety of
(
n
+
1
2
)
{(n+\frac{1}{2})}
-Engel groups lies between the varieties of n-Engel groups and
(
n
+
1
)
{(n+1)}
-Engel groups.
In this paper, we study these groups, and in particular, we prove that all
(
4
+
1
2
)
{(4+\frac{1}{2})}
-Engel
{
2
,
3
}
{\{2,3\}}
-groups are locally nilpotent.
We also show that if G is a
(
4
+
1
2
)
{(4+\frac{1}{2})}
-Engel p-group, where
p
≥
5
{p\geq 5}
is a prime, then
G
p
{G^{p}}
is locally nilpotent.
Title: On (???? + ½)-Engel groups
Description:
Abstract
Let n be a positive integer.
We say that a group G is an
(
n
+
1
2
)
{(n+\frac{1}{2})}
-Engel group if it satisfies the law
[
x
,
y
n
,
x
]
=
1
{[x,{}_{n}y,x]=1}
.
The variety of
(
n
+
1
2
)
{(n+\frac{1}{2})}
-Engel groups lies between the varieties of n-Engel groups and
(
n
+
1
)
{(n+1)}
-Engel groups.
In this paper, we study these groups, and in particular, we prove that all
(
4
+
1
2
)
{(4+\frac{1}{2})}
-Engel
{
2
,
3
}
{\{2,3\}}
-groups are locally nilpotent.
We also show that if G is a
(
4
+
1
2
)
{(4+\frac{1}{2})}
-Engel p-group, where
p
≥
5
{p\geq 5}
is a prime, then
G
p
{G^{p}}
is locally nilpotent.
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