Definition 2

Let G be a totally ordered group. If the number of isolated subgroups of G distinct from G is finite and equal to n, G is said to be of height n. If this number is infinite, G is said to be of infinite height.

Examples (1) The height of the group $G = \{0\}$ is 0. (2) The groups $\mathbf{Z}$ and $\mathbf{R}$ are of height 1. (3) Let $G$ be a totally ordered group and $H$ an isolated subgroup of $G$. If $h(H)$ and $h(G/H)$ denote the heights of the totally ordered groups $H$ and $G/H$, then

since the set of isolated subgroups of $G$ is totally ordered by inclusion. In particular, if $G$ is the lexicographic product of two totally ordered groups $H$ and $H'$, then (cf. no. 2, Example 2); thus the lexicographic product $\mathbf{Z} \times \mathbf{Z}$ is of height 2.

On the other hand the height of $\mathbf{Z} \times \mathbf{Z}$ ordered by embedding in $\mathbf{R}$ (cf. § 3, no. 4, end of Example 6) is equal to 1 (cf. Proposition 8 below).