Isoclinic Galois groups
Two groups \(G\) and \(H\) are called isoclinic if there exist isomorphisms \(\theta: G/Z(G)\to H/Z(H)\) and \(\phi:G'\to H'\) such that \(\phi([x,y])=[x',y']\), where \(x'Z(H)=\theta(xZ(G))\) and \(y'Z(H)=\theta(yZ(G))\). (Here \(Z(G)\) is the centre of \(G\), and \(G'\) is the commutator subgroup of \(G\), generated by all commutators \([x,y]=x^{-1}y^{-1}xy\).) In this paper Schmid gives a different description for stem groups. A group \(X\) is a stem group provided \(Z(X)\subseteq X'\). The author considers the embedding problem in Galois theory related to a central stem extension \(1\to Z\to X\to G\to 1\) of a finite group \(G\). As a corollary of the main result Schmid shows that if \(X\) is realizable as a Galois group over any number field \(k\), then all groups in the isoclinism class of \(X\) are also realizable as Galois groups over \(k\). The other groups appear as Galois groups of subfields of the compositum of certain abelian extensions of \(k\), whose existence can be determined by class field theory. The obtained results can be used in the study of the realizability of \(p\)-groups as Galois groups over number fields.
scientific article; zbMATH DE number 6495600