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I am struggling with the following question.

Suppose I have a group $H$ which is a subgroup of $\mathbb{Z}\oplus\mathbb{Z}$, such that any element $\begin{bmatrix} a \\[0.3em] b \end{bmatrix}$ is defined as: if $b=0$, then $a=0$. How can I prove that $H$ has a basis with exactly one element?

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    A clearer way to state this problem would be to say that $H$ is a subgroup of ${\mathbb Z} \oplus {\mathbb Z}$ having zero intersection with the first direct factor.2012-05-01
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    What is exactly the meaning of a one element basis? I am under the impression that it is a set with only one element in it; like (x). But the answers here suggest that one element basis is of the type (x,y). I am a bit confused with this.2012-05-09
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    This is because elements of $\mathbb{Z}\oplus \mathbb{Z}$ are pairs $(x,y)$ with $x,y\in \mathbb{Z}$.2012-05-09
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    @CasterT: The comment above is addressed to you.2012-05-09

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