Baer invariant

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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).