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# Free Product

TOPO-ELKZVK

Which of the following statements is true about $\mathbb Z*\mathbb Z=\langle a,b\rangle$?

A

$\mathbb Z*\mathbb Z$ has nontrivial center.

B

There exists a homomorphism $\varphi:\mathbb Z*\mathbb Z\rightarrow\mathbb Z\times\mathbb Z$ with the following universal property: for any abelian group $A$ and homomorphism $f:\mathbb Z*\mathbb Z\rightarrow A$ there exists a unique homomorphism $\tilde{f}:\mathbb Z\times\mathbb Z\rightarrow A$ such that $f=\tilde{f}\circ\varphi$.

C

The kernel of any homomorphism $\mathbb Z*\mathbb Z\rightarrow\mathbb Z\times\mathbb Z$ has infinite rank.

D

The commutator subgroup of $\mathbb Z*\mathbb Z$ is cyclic.

E

$\mathbb Z*\mathbb Z$ has a nontrivial finite subgroup.