It is shown that over an arbitrary field there exists a nil algebra
R
R
whose adjoint group
R
o
R^{o}
is not an Engel group. This answers a question by Amberg and Sysak from 1997. The case of an uncountable field also answers a recent question by Zelmanov.
In 2007, Rump introduced braces and radical chains
A
n
+
1
=
A
⋅
A
n
A^{n+1}=A\cdot A^{n}
and
A
(
n
+
1
)
=
A
(
n
)
⋅
A
A^{(n+1)}=A^{(n)}\cdot A
of a brace
A
A
. We show that the adjoint group
A
o
A^{o}
of a finite right brace is a nilpotent group if and only if
A
(
n
)
=
0
A^{(n)}=0
for some
n
n
. We also show that the adjoint group
A
o
A^{o}
of a finite left brace
A
A
is a nilpotent group if and only if
A
n
=
0
A^{n}=0
for some
n
n
. Moreover, if
A
A
is a finite brace whose adjoint group
A
o
A^{o}
is nilpotent, then
A
A
is the direct sum of braces whose cardinalities are powers of prime numbers. Notice that
A
o
A^{o}
is sometimes called the multiplicative group of a brace
A
A
. We also introduce a chain of ideals
A
[
n
]
A^{[n]}
of a left brace
A
A
and then use it to investigate braces which satisfy
A
n
=
0
A^{n}=0
and
A
(
m
)
=
0
A^{(m)}=0
for some
m
,
n
m, n
.
We also describe connections between our results and braided groups and the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. It is worth noticing that by a result of Gateva-Ivanova braces are in one-to-one correspondence with braided groups with involutive braiding operators.