order of a profinite group

Let $G$ be a profinite group, and let $H$ be any closed subgroup. We define the of $H$ in $G$ by

$[G:H]=\mathrm{lcm}(\{[G/N:HN/N]\}),$

where $N$ runs over all open (and hence of finite index) subgroups of $G$, and where $\mathrm{lcm}$ is taken in the sense of the least common multiple of supernatural numbers.

In particular, we can define the order of a profinite group to be the index of the identity subgroup in $G$:

$|G|:=[G:\{e\}].$

Some examples of orders of profinite groups:

• $G={\mathbb{Z}}_p$, the ring of $p$-adic integers. Since every finite quotient of ${\mathbb{Z}}_p$ is cyclic of $p^n$ elements (for some $n$), and every such group occurs as a quotient, we have $|G|=\mathrm{lcm}(p^n),$ where $n$ runs over all natural numbers. Thus $|G|=p^{\mathrm{\infty }}$.

• $G=\widehat{\mathbb{Z}}.$ Since $G\approx {\prod }_p{\mathbb{Z}}_p$, we have $|G|={\prod }_p|{\mathbb{Z}}_p|={\prod }_p{p}^{\mathrm{\infty }}$. This example illustrates the limitations of this concept: Despite being “relatively small” in of profinite groups, $\widehat{\mathbb{Z}}$ has the largest possible profinite order.