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# 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 abelian group, can be extended to a group homomorphism $\overline{\rho}:A\rightarrow H$.

Which of the following are FALSE?

Select ALL that apply.

A

The elements of every free abelian group satisfy no non-trivial relations.

B

Every group is a quotient of a free group.

C

A free abelian group with $n\ge 2$ basis elements is a free group.

D

If a free abelian group is a free group then it is cyclic.

E

The elements of a free group satisfy no non-trivial relations.