Direct product of group of rational numbers and group of rational numbers modulo integers

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Definition

This group can be defined in the following equivalent ways:

1. It is the external direct product of the group of rational numbers $\mathbb {Q}$ and the group of rational numbers modulo integers $\mathbb {Q} /\mathbb {Z}$. Explicitly, it is the group $\mathbb {Q} \times \mathbb {Q} /\mathbb {Z}$.
2. It is the external central product of two copies of the group of rational numbers where we identify a copy of the group of integers in group of rational numbers in both groups with each other. Explicitly, it is the central product $\mathbb {Q} *_{\mathbb {Z} }\mathbb {Q}$.

Related groups

• Amalgamated free product of two copies of group of rational numbers over group of integers