Nil group
A group in which any two elements and
are connected by a relation
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where the square brackets denote the commutator
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and the number of commutators in the definition depends, generally speaking, on the pair
. When
is bounded for all
in the group, the group is called an Engel group. Every locally nilpotent group is a nil group. The converse is not true, in general, but it is under some additional assumptions, for example, when the group is locally solvable (cf. Locally solvable group).
Occasionally the term "nil group" is used in a different meaning. Namely, a nil group is a group in which every cyclic subgroup is subnormal, that is, occurs in some subnormal series of the group (see Normal series of a group).
References
[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
Comments
In [a1] it has been proved that there are periodic Engel groups that are not locally nilpotent.
References
[a1] | E.S. Golod, "On nil-algebras and residually finite ![]() |
Nil group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_group&oldid=31721