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Abstract Algebra

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Free Abelian Groups: $A=B+3A$, $A$ Free Abelian, $B$ a Subgroup

ABSALG-CH94BP

Let $A$ be the abelian product group $\mathbb{Z}\times \mathbb{Z}$.

Suppose that $B$ is a subgroup of $A$ with $A=B+3A$.

Select "True" or "False" for each of the following statements.

True

False

True

False

The map $p:A\rightarrow A/3A$, where $p(a)=a+3A$, $a\in A$, satisfies $p(B)=A/3A$.

True

False

The group $B$ is a cyclic subgroup of $A$.

True

False

The index $[A:B]$ is finite.

True

False

$3$ divides $[A:B]$.