# History of profinite groups, when was it first mentioned? What was the original definition?

Searching left me hanging. One of my professors told me the definition using the topological properties was the first one but I cannot find any resources. Is that true? If not, how was it originally defined? References would be lovely. Best regards

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Profinite groups were first called "Groups of Galois type", see J.P. Serre's book "Cohomologie Galoisienne" of $1964$. The term "profinite" comes from Serre (if I am not mistaken). Of course, some profinite groups have a much older history, e.g., already Hensel defined in $1910$ the $p$-adic integers during his studies of algebraic numbers.

As to the definition, a profinite group is a Hausdorff, compact, and totally disconnected topological group. The other (equivalent) definition, better adapted to the name "profinite", is, that a profinite group is a group which is isomorphic to the inverse limit of an inverse system of discrete finite groups.

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Thanks for the answer. I'll look into that –  Horstenson Jul 15 '13 at 19:20
So is the theorem that a topological group is isomorphic to an inverse limit of finite discrete groups iff it is compact and totally disconnected also due to Serre? –  Pete L. Clark Jul 15 '13 at 22:34
Interesting history point, but the result is implicit in the Peter-Weyl theorem? Whose application to compact topological groups was left implicit for a decade or so? I was once left pondering the remark of Dennis Sullivan that compact groups are the inverse limits of compact Lie groups, which is probably similarly folkloric. –  Charles Matthews Jul 16 '13 at 8:18
Perhaps more can be found in the letters of the correspondence between Serre and Grothendieck (1955-1957), where they discuss (among other things) inductive and inverse limits, "topological groups" and "homological algebra". Grothendieck writes: "some of this theorems have been proved umpteen times in all kinds of special cases". –  Dietrich Burde Jul 16 '13 at 8:56