Abelian groups

Definition: Abelian group

A group $G$ is abelian if the group operation commutes, meaning $ab=ba$ for all $a,b\in G$.

Related notes:

• Group generators, relations, and presentations
• Commutator subgroups and abelianization

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Free abelian group

Definition: Free abelian group

The free abelian group with basis $X$ is the group of formal linear combinations of elements of $X$ with coefficients in $\mathbb{Z}$: $\mathbb{Z}X=\left\{{\sum }_{x\in X}{n}_x\cdot x:n_x=0\in Z\text{ for all but finitely many }\sigma \right\}.$

Lemma: Homomorphisms and the free abelian group

Let $X$ be any space and $A$ be an abelian group.

• (i) Two group homomorphisms $\phi ,\psi :\mathbb{Z}X\to A$ out of the free abelian group agree if and only if $\phi (x)=\psi (x)$ for all $x\in X$.
• (ii) For any function $f:X\to A$, there exists a homomorphism $\phi :\mathbb{Z}X\to A$ with $\phi (x)=f(x)$ for all $x\in X$.

In the language of categories, this is the claim that the function $\mathsf{Ab}(\mathbb{Z}X,A)\to \mathsf{Sets}(X,A)$ sending $\phi \to \phi {|}_X$ is a bijection for any set $X$ and abelian group $A$.

Proof from Algebraic Topology.

• (i) Recall that homomorphisms preserve linear combinations, so two homomorphisms agree on $X$ if and only if they agree on the $\mathbb{Z}$-linear combination of elements of $X$, i.e., $\mathbb{Z}X$.
• (ii) For a function $f:X\to A$, we can define $\phi :\mathbb{Z}X\to A$ by $\phi (c)={\sum }_{x\in X}{c}_{x}f(x)$ for $c=\sum c_{x}x$. The map $\phi$ is well-defined because the terms $c_x$ are uniquely determined by $c$, and we clearly have $\phi (x)=f(x)$ for all $x\in X$. Finally, $\phi$ is a homomorphism by definition of addition in $\mathbb{Z}X$. $\square$