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 extensions

Consider possible group extensions of the form:

$1\to A\to E\to G\to 1$

satisfying the condition that the image of $A$ in $E$ is contained in the ${\mathcal {V}}$ -marginal subgroup of $E$ . Consider the set of defining words for ${\mathcal {V}}$ (note that it suffices to take any generating set of words for the variety). For each word $w$ with $n_{w}$ letters, we have a word map $E^{n_{w}}\to E$ (a set map only, not a homomorphism). There is a natural initial object dependent only on $G$ and ${\mathcal {V}}$ that has a unique homomorphism to the ${\mathcal {V}}$ -verbal subgroup of $E$ . This natural initial objection, which we will call $V^{\#}(G)$ , has a unique homomorphism to the ${\mathcal {V}}$ -verbal subgroup $V(G)$ . The kernel of this homomorphism is the Baer invariant of $G$ . PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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 $c$	${\frac {R\cap \gamma _{c+1}(F)}{[[\dots [R,F],F],\dots ],F]}}$ ( $F$ occurs $c$ times in the denominator)	also called the $c$ -nilpotent multiplier and denoted $M^{(c)}(G)$ .