# Baer invariant

## Contents

## Definition

Suppose is a group. Suppose is a subvariety of the variety of groups (note that may or may not be in ). The **Baer invariant** of with respect to , denoted , is an abelian group defined as follows.

### Definition in terms of extensions

Consider possible group extensions of the form:

satisfying the condition that the image of in is contained in the -marginal subgroup of . Consider the set of defining words for (note that it suffices to take any generating set of words for the variety). For each word with letters, we have a word map (a set map only, not a homomorphism). There is a natural initial object dependent only on and that has a unique homomorphism to the -verbal subgroup of . This natural initial objection, which we will call , has a unique homomorphism to the -verbal subgroup . The kernel of this homomorphism is the Baer invariant of .

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### Definition in terms of presentation

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

Denote by the verbal subgroup of corresponding to all the words defining the variety , so is the largest quotient of that is in . Also, define as the subgroup generated by all words of the form:

where varies over all words defining the variety , (and are also free to vary) and varies over all of .

Then, the **Baer invariant** of with respect to is defined as:

## Particular cases

Case on subvariety | Description of Baer invariant using presentation | Other names and comments |
---|---|---|

abelian groups | also called the Schur multiplier and denoted . See Hopf's formula for Schur multiplier. | |

nilpotent groups of class at most | ( occurs times in the denominator) | also called the -nilpotent multiplier and denoted . |