Notes on the minimal number of ramified primes in some \(l\)-extensions of \(\mathbb Q\) (Q933729)

One of the issues studied in the context of the Galois inverse problem is that for a rational prime number \(l\) and a given finite \(l\)--group \(G\) of order \(l^ n\), find the minimal number \(\text{ram}^ t (G)\) such that \(G\) can be realized as the Galois group of an extension \(K\) of \({\mathbb Q}\) ramified at only \(\text{ram}^ t(G)\) rational primes. If \(d(G)\) denotes the minimal number of generators of \(G\) we have \(d(G) \leq \text{ram}^ t(G)\leq n\). The main purpose of the paper is to prove that for any \(3\)--group \(G\) of order less than of equal to \(3^ 5\), \(\text{ram}^ t(G) =d(G)\). This result is a consequence of the following result proved in the paper: If \(l\) is odd, \(H=G/(Z(G)\cap G')\) where \(Z(G)\) is the center of \(G\) and \(G'\) is the commutator subgroup, then \(\text{ram}^ t(G)\leq d(G)+\sum_{2\leq i\leq m} d_ i (H)\) where \(d_ i(H)\) is the minimal number of generators of \(C_ i(H)/ C_ {i+1}(H)\) and \(C_ i(H)\) denotes the \(i\)--th commutator subgroup of \(H\) and \(m\) the nilpotency class of \(H\).