This is probably well-known:


Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed?


~~- Is every profinite group isomorphic to the Galois group of some Galois extension?~~ (yes, see the comments)



profinite spaces coming from profinite groups
(Note: This was intended to be a comment to unknown (google)'s answer - but as I'm new here I can't post comments.)


As Pete L. Clark points out, unknown (google)'s answer is false as stated.
However, this is only because of the omission of the word "infinite".
A correct statement is:


An infinite profinite group $G$ is homeomorphic to $\{0,1\}^{w(G)}$,
where $\{0,1\}$ is the $2$-point discrete space,
and $w(G)$ is the weight of $G$.


This is Theorem 9.15 (pages 95-98) of "Abstract Harmonic Analysis I" by Edwin Hewitt and Kenneth A. Ross. (Hewitt and Ross actually state the result using the minimum cardinality of a local base at $1\_G$, rather than the weight of $G$, but these are equal for infinite profinite groups.)


Notice that the case of countable weight is an immediate consequence of the usual characterisation of the Cantor set.


