NilpotentEngelGroups

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

 The commutator x-1y-1xy of x and y is denoted by [x,y]. It satisfies the identity xy = yx [x,y]. The iterated commutator [x, n+1y] is defined recursively by [x, 1y] = [x,y] [x, n+1y] = [[x, ny], 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, nh] = 1 in G. G is called an n-Engel group (or Engel-n group) if there is a positive integer n such that [g, nh] = 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, ny] 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 Z2, Z, Z2, Z3,
C2×Z2,
Z
E(3,4) 9 88 C544× C105× C304× C604 Z3, Z3, Z8, Z18,
Z24× C24× C105× C30,
Z26× C59× C109,
Z6× C523× C10× C303,
C53× C303,
C3
E(2,5) 9 23 C38× C303× C1802 Z2, Z, Z2, Z3, Z6,
C2×C6× Z4,
C62× C182× Z4,
C2×C303× Z,
C34× C152
E(2,6) 12 70 C75× C1415× C8410×
C1683× C8402× C2520×
C126003× C321564600 2
Z2, Z, Z2, Z3, Z6, Z9,
C2× C62× Z12,
C23× C422× C843× Z13,
C29× C426× C1262× Z14,
C213× C143× C428× C1050× C63002× Z8,
C27× C1410× C2102× C53594100 2,
C22× C10
E(2,7) > 11 > 149