Baer invariant

Definition

Suppose $G$ is a group. Suppose ${\mathcal {V}}$ is a subvariety of the variety of groups (note that $G$ may or may not be in ${\mathcal {V}}$). The Baer invariant of $G$ with respect to ${\mathcal {V}}$, denoted ${\mathcal {V}}M(G)$, is an abelian group defined as follows.

Definition in terms of presentation

Suppose $G$ is expressed in the form $F/R$ where $F$ is a free group and $R$ is the normal closure of a set of words in $F$. Explicitly, any presentation of $G$ can be viewed in this manner, where $F$ is the free group on symbols corresponding to the generators and $R$ is the subgroup obtained as the normal closure of the relation words.

Denote by $V(F)$ the verbal subgroup of $F$ corresponding to all the words defining the variety ${\mathcal {V}}$, so $F/V(F)$ is the largest quotient of $F$ that is in ${\mathcal {V}}$. Also, define $V^{*}(R,F)$ as the subgroup generated by all words of the form:

$v(f_{1},f_{2},\dots ,f_{i}r,f_{i+1},\dots ,f_{n})v(f_{1},f_{2},\dots ,f_{n})^{-1}$

where $v$ varies over all words defining the variety ${\mathcal {V}}$, $f_{1},f_{2},\dots ,f_{n}\in F$ (and $n,i$ are also free to vary) and $r$ varies over all of $R$.

Then, the Baer invariant of $G$ with respect to ${\mathcal {V}}$ is defined as:

${\mathcal {V}}M(G)={\frac {R\cap V(F)}{V^{*}(R,F)}}$

Particular cases

Case on subvariety	Description of Baer invariant using presentation	Other names and comments
abelian groups	${\frac {R\cap [F,F]}{[R,F]}}$	also called the Schur multiplier and denoted $M(G)$. See Hopf's formula for Schur multiplier.
nilpotent groups of class at most $c$	Failed to parse (syntax error): {\displaystyle \frac{R \cap \gamma_c(F)}{[[ \dots[R,F],F],\dots],F]}	also called the $c$-nilpotent multiplier and denoted $M^{(c)}(G)$.