Engel group

A group $G$ in which for every two elements $a,b\in G$ there is an integer $n=n(a,b)$ such that $[[\ldots[[a,b],b],\ldots],b]=1$, where $b$ is separated $n$ times and $[a,b]$ is the commutator of $a$ and $b$. If this number $n$ can be chosen independently of $a$ and $b$, then $G$ is called an Engel group of finite class $n$. The class of Engel groups contains that of locally nilpotent groups, but does not coincide with it. Every nilpotent group of class $n$ is an Engel group of the same class. Engel groups of class 2 are nilpotent of class at most 3.

They are named after F. Engel.

References

[1]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)

Comments

A finite Engel group is nilpotent (cf. Nilpotent group).

References

[a1]	D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
[a2]	B. Huppert, "Finite groups" , 3 , Springer (1982)