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



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


