What is an isolated subgroup?

Question

" As in the case of finite numbers, the infinitesimal numbers form an isolated subgroup  of $R_{inf}$ of $^*R$"page 152

What does this sentence mean? What is an isolated subgroup?

Answer 1

The full quote reads:

As in the case of finite numbers, the infinitesimal numbers form an isolated subgroup $\mathbf{R}_{\text{inf}}$ of $^*\mathbf{R}$; this implies that the factor group $^*\mathbf{R}/\mathbf{R}_{\text{inf}}$ inherits naturally its order relation from $^*\mathbf{R}$.

From this context, I would infer that for an ordered group $(G,\leq)$, a subgroup $H$ is isolated if for all $x,y\in H$, if $z\in G$ with $x\leq z \leq y$, then $z\in H$. This condition is necessary and sufficient for the ordering $\leq$ on $G$ to induce an ordering on the quotient group $G/H$. And the infinitesimal numbers certainly satisfy this condition.

I have never heard the term "isolated" used in this way - I would call such a subgroup $H$ "convex". The very first Google result for "isolated subgroup" is this page: https://encyclopediaofmath.org/wiki/Isolated_subgroup. Here Encyclopedia of Math gives a different (non-order-theoretic) definition. But it also notes

In the theory of ordered groups, isolated subgroups are sometimes referred to as convex subgroups.

This is a bit odd: it's clear that a convex subgroup of an ordered group is isolated (by the EOM definition), but it seems to me that there are isolated subgroups of ordered groups which are not convex (e.g. $\mathbb{Q}\subseteq \mathbb{R}$ as additive groups). But in any case, it seems relatively clear that "isolated" means "convex" in the context of the paper you're reading.