# 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:

$v(f_1,f_2,\dots,f_ir,f_{i+1},\dots,f_n)v(f_1,f_2,\dots,f_n)^{-1}$

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