Engel group
From Encyclopedia of Mathematics
A group in which for every two elements
there is an integer
such that
, where
is separated
times and
is the commutator of
and
. If this number
can be chosen independently of
and
, then
is called an Engel group of finite class
. The class of Engel groups contains that of locally nilpotent groups, but does not coincide with it. Every nilpotent group of class
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) |
How to Cite This Entry:
Engel group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Engel_group&oldid=31929
Engel group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Engel_group&oldid=31929
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article