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.
References
 1 [Ram] Ramakrishnan, Dinikara and Valenza, Robert. Fourier Analysis on Number Fields. Graduate Texts in Mathematics, volume 186. SpringerVerlag, New York, NY. 1989.
 2 [Ser] Serre, J.P. (Ion, P., translator) Galois Cohomology. Springer, New York, NY. 1997
Title  order of a profinite group 

Canonical name  OrderOfAProfiniteGroup 
Date of creation  20130322 15:23:39 
Last modified on  20130322 15:23:39 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  5 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 20E18 
Defines  index of a profinite subgroup 
Defines  index of a profinite group 