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

$G$

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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 $M^{(c)}(G)$.