In my own research I use profinite groups quite frequently (for Galois groups and etale fundamental groups). However my use of them amounts to book-keeping: I only care about finite levels (finite Galois extensions; finite covers) and so I take their inverse limit. Then various "topological" arguments just mean: look at the finite levels (this occurs because the quotients by the open subgroups exactly correspond to the finite levels I care about.)

There are people who use profinite groups more intently than I do, and I suspect that they see some value in them as mathematical objects. So my question is this:

### Question

What theorems/properties are good about profinite groups that *don't* arise trivially from its book-keeping nature (so for example the profinite Sylow theorems are disqualified, because they arise trivially from the finite-level group theory). What, if anything, rewards us for dealing with this new type of object rather than with the finite levels? (again, except that it makes it easier to write down notes.)

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helps. $\endgroup$ – LSpice Feb 14 '18 at 21:03