Nilpotent multiplier
From Groupprops
Definition
Suppose is a positive integer. The
-nilpotent multiplier of a group
, denoted
is defined as the Baer invariant of
with respect to the variety of groups of nilpotency class (at most)
. If we write
where
is a free group, this can be written as:
where denotes the
member of the lower central series of
and the denominator group has
occurrences of
.
Particular cases
In the case , we get the Schur multiplier.
Particular cases based on group property
Group property | Description of nilpotent multipliers |
---|---|
abelian group | nilpotent multiplier of abelian group is graded component of free Lie ring |
perfect group | nilpotent multiplier of perfect group equals Schur multiplier |