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

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

• $G=\mathbb{Z}_p$ , 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 terms of profinite groups, $\widehat{\mathbb{Z}}$ has the largest possible profinite order.

Bibliography

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