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


where N runs over all open (and hence of finite index) subgroupsMathworldPlanetmathPlanetmath of G, and where lcm is taken in the sense of the least common multipleMathworldPlanetmathPlanetmath of supernatural numbers.

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


Some examples of orders of profinite groups:

  • G=p, the ring of p-adic integers. Since every finite quotientPlanetmathPlanetmath of p is cyclic of pn elements (for some n), and every such group occurs as a quotient, we have |G|=lcm(pn), where n runs over all natural numbersMathworldPlanetmath. Thus |G|=p.

  • G=^. Since Gpp, we have |G|=p|p|=pp. This example illustrates the limitations of this concept: Despite being “relatively small” in of profinite groups, ^ has the largest possible profinite order.


  • 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