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 .
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|