Frattini covers of profinite groups (Q797685)

For a profinite group \(G\), its universal Frattini cover or projective cover consists of a projective profinite group \(\tilde G\) and an epimorphism \(\gamma\colon\widetilde G\to G\) with \(\ker\gamma\) contained in the Frattini subgroup of \(\tilde G\). Such covers appear, for example, in the study of absolute Galois groups of pseudo-algebraically closed fields. This paper describes the structure of \(\tilde G\) for certain groups \(G\). A general result is that the finite (continuous) homomorphic images of \(\tilde G\) are completely determined by those of \(G\). If \(G\) is a semidirect product of subgroups \(K\) and \(H\) of relatively prime orders, then \(\tilde G\) is a semidirect product of \(\tilde K\) and \(\tilde H\). The main result of the paper describes the structure of \(\tilde G\) for a group \(G\) that admits a subnormal series \(G=G_ 0\triangleright G_ 1\triangleright\cdots\triangleright G_ n=1\) with each \(G_{i-1}/G_ i\) a finitely generated pro-\(p\)-group, for different primes \(p_ 1,...,p_ n\). In this case \(\tilde G\) has an analogous structure, with its \(p\)-Sylow subgroups free pro-\(p_ i\) of known rank.