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Consider the set on nonzero-rational numbers $\mathbb{Q}^*= \mathbb{Q}-\{0\}$ as a subgroup of the nonzero real numbers $\mathbb{R}^*=\mathbb{R}-\{0\},$ where the group operation $\cdot$ is multiplication. As a subgroup, $\mathbb{Q}^*$ has the additional property that:

  1. $\forall x\in \mathbb{R}^* \forall y\in \mathbb{Q}^*(x\cdot y\in \mathbb{Q}^*\Rightarrow x\in \mathbb{Q}) $ .

Can you give me the name for subgroups obeying property (1) please?

(edit: previously I had listed an additional property which was incorrect, which I since deleted.)

Arturo Magidin
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Eoin
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1 Answers1

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The property is "being a subgroup": every subgroup of a group has that property.

Let $G$ be a group; let $H$ be a subgroup of $G$. If $x\in G$ and $y\in H$ are such that $xy\in H$, then $x\in H$. Indeed, $x=(xy)y^{-1}\in H$, since $H$ is closed under products and inverses; hence $y^{-1}\in H$, and $xy,y^{-1}\in H$ implies their product is in $H$.

It is also the case that $yx\in H$ implies $x\in H$.

As an alternative argument: if $xy\in H$, then $xH\cap H\neq\varnothing$, and therefore $xH=H$; but this holds if and only if $x\in H$.

Arturo Magidin
  • 398,050