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

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

where $N$ runs over all open (and hence of finite index) subgroups of $G$ , and where $\operatorname{lcm}$ 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 $G$ :

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

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 $|G|=\operatorname{lcm}(p^{n}),$ where $n$ runs over all natural numbers. Thus $|G|=p^{\infty}$ .
•

$G=\widehat{\mathbb{Z}}.$ Since $G\approx\prod_{p}\mathbb{Z}_{p}$ , we have $|G|=\prod_{p}|\mathbb{Z}_{p}|=\prod_{p}p^{\infty}$ . 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. Springer-Verlag, 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	2013-03-22 15:23:39
Last modified on	2013-03-22 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