Nilpotent multiplier

Definition

Suppose $c$ is a positive integer. The $c$-nilpotent multiplier of a group $G$, denoted $M^{(c)}(G)$ is defined as the Baer invariant of $G$ with respect to the variety of groups of nilpotency class (at most) $c$. If we write $G=F/R$ where $F$ is a free group, this can be written as:

$M^{(c)}(G)=\frac{R\cap {\gamma }_{c+1}(F)}{[[\dots [R,F],\dots ,],F]}$

where ${\gamma }_{c+1}(F)$ denotes the $(c+1{)}^{th}$ member of the lower central series of $F$ and the denominator group has $c$ occurrences of $F$.

Particular cases

In the case $c=1$, we get the Schur multiplier.