Hopf's formula for nilpotent multiplier

Statement

Let $G$ be a group isomorphic to the quotient group $F/R$, where $F$ is a free group and $R$ is a normal subgroup of $F$. Suppose $c$ is a positive integer. Then, the $c$-nilpotent multiplier of $G$, denoted $M^{(c)}(G)$, is an abelian group given by the formula:

$M^{(c)}(G)\cong (R\cap \gamma _{c+1}(F))/\gamma _{c+1}(R,F)$.

Here:

• $\gamma _{c+1}(F)$ is the $(c+1)^{th}$ member of the lower central series of $F$, which is described explicitly as a $(c+1)$-fold iterated commutator of copie of $F$. Inductively, it is defined as $\gamma _{1}(F)=F$ and $\gamma _{i+1}(F)=[F,\gamma _{i}(F)]$
• $\gamma _{c+1}(R,F)$ is defined as the $(c+1)^{th}$ member of the series defined inductively as $\gamma _{1}(R,F)=R$ and $\gamma _{i+1}(R,F)=[F,\gamma _{i}(R,F)]$

Note that any choice of generating set for $G$ gives a choice of $F$ and $R$ for which the theorem can be applied: $F$ is the free group on those generators with the natural surjection, and $R$ is the kernel of the surjection.

Related facts

• Hopf's formula for Baer invariant
• Hopf's formula for Schur multiplier: This is the special case $c=1$.

Applications

• Formula for nilpotent multiplier of free nilpotent group