Nilpotent
Engel Groups |

Let G be a group and x,y elements of G.

The **commutator** x^{-1}y^{-1}xy of
x and y is denoted by [x,y]. It satisfies the identity xy = yx [x,y].
The iterated commutator [x, _{n+1}y] is defined
recursively by
_{1}y] = [x,y]

[x, _{n+1}y] = [[x, _{n}y], y].
G is called an **Engel group** if for all elements g
and h in G there is a positive integer n (which depends on g an h)
such that [g, _{n}h] = 1 in G.
G is called an **n-Engel group** (or **Engel-n
group**) if there is a positive integer n such that
[g, _{n}h] = 1 in G for all elements g and h
in G.
The **free nilpotent d-generator Engel-n group** is
denoted by **E(d,n)**. Since [x, _{n}y]
is an element of the (n+1)th term of the lower central series, the
class-n quotient of E(d,n) is the free nilpotent group of rank d and
class n.

class
Hirsch length
torsion

subgroup
lower central

series
E(2,4)
6
11
1
**Z**^{2}, **Z**,
**Z**^{2}, **Z**^{3},

C_{2}×**Z**^{2},

**Z**
E(3,4)
9
88
C _{5}^{44}×
C_{10}^{5}×
C_{30}^{4}×
C_{60}^{4}
**Z**^{3},
**Z**^{3},
**Z**^{8},
**Z**^{18},

**Z**^{24}×
C_{2}^{4}×
C_{10}^{5}×
C_{30},

**Z**^{26}×
C_{5}^{9}×
C_{10}^{9},

**Z**^{6}×
C_{5}^{23}×
C_{10}×
C_{30}^{3},

C_{5}^{3}×
C_{30}^{3},

C_{3}
E(2,5)
9
23
C _{3}^{8}×
C_{30}^{3}×
C_{180}^{2}
**Z**^{2},
**Z**,
**Z**^{2},
**Z**^{3},
**Z**^{6},

C_{2}×C_{6}×
**Z**^{4},

C_{6}^{2}×
C_{18}^{2}×
**Z**^{4},

C_{2}×C_{30}^{3}×
**Z**,

C_{3}^{4}×
C_{15}^{2}
E(2,6)
12
70
C _{7}^{5}×
C_{14}^{15}×
C_{84}^{10}×

C_{168}^{3}×
C_{840}^{2}×
C_{2520}×

C_{12600}^{3}×
C_{321564600}
^{2}
**Z**^{2},
**Z**,
**Z**^{2},
**Z**^{3},
**Z**^{6},
**Z**^{9},

C_{2}×
C_{6}^{2}×
**Z**^{12},

C_{2}^{3}×
C_{42}^{2}×
C_{84}^{3}×
**Z**^{13},

C_{2}^{9}×
C_{42}^{6}×
C_{126}^{2}×
**Z**^{14},

C_{2}^{13}×
C_{14}^{3}×
C_{42}^{8}×
C_{1050}×
C_{6300}^{2}×
**Z**^{8},

C_{2}^{7}×
C_{14}^{10}×
C_{210}^{2}×
C_{53594100
}^{2},

C_{2}^{2}×
C_{10}
E(2,7)
> 11
> 149