Easy# Free Group versus Free Abelian Group: Definitions and Properties

ABSALG-XLLNNT

Recall one definition of free group: Let $F$ be a group and $B$ a subset of $F$. Then $F$ is a ** free group** with basis $B$ if every map $\varphi:B\rightarrow G$, where $G$ is a group, can be extended to a group homomorphism $\overline{\varphi}:F\rightarrow G$. (If $B=\emptyset$, then $F=\{e\}$, where $e$ is the identity element.)

Compare this to the definition of a free *abelian* group: Let $A$ be an *abelian* group and $S$ a subset of $A$. Then $A$ is a ** free abelian group** with basis $S$ if every map $\rho: S\rightarrow H$, where $H$ is an

Which of the following are **FALSE**?

Select **ALL** that apply.