group theory

# Contents

## Idea

The free abelian group $ℤ\left[S\right]$ on a set $S$ is the abelian group whose elements are formal $ℤ$-linear combinations of elements of $S$.

## Definition

Let

$U:\mathrm{Ab}\to \mathrm{Set}$U : Ab \to Set

be the forgetful functor from the category Ab of abelian groups, to the category Set of sets. This has a left adjoint free construction:

$ℤ\left[-\right]:\mathrm{Set}\to \mathrm{Ab}\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{Z}[-] : Set \to Ab \,.
###### Definition

This is the free abelian group functor. For $S\in$ Set, the free abelian group $ℤ\left[S\right]\in$ Ab is the free object on $S$ with respect to this free/forgetful adjunction.

Explicit descriptions of free abelian groups are discussed below.

## Properties

### In terms of formal linear combinations

###### Definition

A formal linear combination of elements of a set $S$ is a function

$a:S\to ℤ$a : S \to \mathbb{Z}

such that only finitely many of the values ${a}_{s}\in ℤ$ are non-zero.

Identifying an element $s\in S$ with the function $S\to ℤ$ which sends $s$ to $1\in ℤ$ and all other elements to 0, this is written as

$a=\sum _{s\in S}{a}_{s}\cdot s\phantom{\rule{thinmathspace}{0ex}}.$a = \sum_{s \in S} a_s \cdot s \,.

In this expression one calls ${a}_{s}\in ℤ$ the coefficient of $s$ in the formal linear combination.

###### Definition

For $S\in$ Set, the group of formal linear combinations $ℤ\left[S\right]$ is the group whose underlying set is that of formal linear combinations, def. 2, and whose group operation is the pointwise addition in $ℤ$:

$\left(\sum _{s\in S}{a}_{s}\cdot s\right)+\left(\sum _{s\in S}{b}_{s}\cdot s\right)=\sum _{s\in S}\left({a}_{s}+{b}_{s}\right)\cdot s\phantom{\rule{thinmathspace}{0ex}}.$(\sum_{s \in S} a_s \cdot s) + (\sum_{s \in S} b_s \cdot s) = \sum_{s \in S} (a_s + b_s) \cdot s \,.
###### Proposition

The free abelian group on $S\in \mathrm{Set}$ is, up to isomorphism, the group of formal linear combinations, def. 3, on $S$.

### In terms of direct sums

###### Proposition

For $S$ a set, the free abelian group $ℤ\left[S\right]$ is the direct sum in Ab of $\mid S\mid$-copies of $ℤ$ with itself:

$ℤ\left[S\right]\simeq {\oplus }_{s\in S}ℤ\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{Z}[S] \simeq \oplus_{s \in S} \mathbb{Z} \,.

### Relation to formal linear combinations with coefficients

The definition 2 of formal linear combinations makes sense with coefficients in any abelian group $A$, not necessarily the integers.

$A\left[S\right]≔ℤ\left[S\right]\otimes A\phantom{\rule{thinmathspace}{0ex}}.$A[S] \coloneqq \mathbb{Z}[S] \otimes A \,.

## Examples

• The free abelian group on the singular simplicial complex of a topological space $X$ consists of the singular chains on $X$.

• For $R$ a ring and $S$ a set, the tensor product of abelian groups $ℤ\left[S\right]\otimes R$ is the free module over $R$ on the basis $S$. If $R=k$ is a field, then this is the vector space over $k$ with basis $S$.

• For $R$ a ring, the tensor product of abelian groups $ℤ\left[ℕ\right]\otimes R$ is the abelian group underlying the ring of polynomials over $R$.

Revised on September 6, 2012 14:49:41 by Urs Schreiber (131.174.188.45)