# 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 OrderOfAProfiniteGroup 2013-03-22 15:23:39 2013-03-22 15:23:39 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 20E18 index of a profinite subgroup index of a profinite group