On Hopf Galois structures and complete groups
The topic of this paper is counting the Hopf Galois structures on a given \(\Gamma\)-Galois extension \(L/K\). It is known that \(L/K\) may be \(H\)-Galois for Hopf algebras \(H\) which are not group rings (work of Pareigis and the reviewer); all eligible \(H\) are ``split by \(L\)'', that is, \(LH\) is a group ring \(L[G]\), but the nice feature which keeps one in business is that the group \(G\) need not be \(\Gamma\). There are results that for certain groups \(\Gamma\), \(G\) must be \(\Gamma\) (e.g. if \(\Gamma\) is nonabelian simple (Byott)). This paper produces examples of the opposite behavior: if \(\Gamma\) is the holomorph of the group of prime order \(p\) (so \(\Gamma\) has order \(p(p-1)\)), and \(q=(p-1)/2\) is again prime, then there are six groups \(G\) of order \(p(p-1)\), and all six of them do occur as \(G\). (Remark on terminology: If \(p\) is a prime such that \(q\) is also prime, then \(p\) is called a safeprime and \(q\) is called a Sophie Germain prime. The reviewer will permit himself the personal remark that he prefers the latter terminology.) The extension \(L/K\) in this case is metabelian of a type that occurs frequently: take for instance \(K=\mathbb Q\) and \(L\) the Galois closure of \({\mathbb Q}({\root p \of x})\), provided this has degree \(p\) over \(\mathbb Q\). So there is a certain intrinsic interest in these extensions. The paper even determines how often each group \(G\) occurs in a precise sense. The approach is based on the correspondence of Greither-Pareigis, in its improved form due to Byott, which reduces everything to pure group theory, and proceeds via systematic (essentially elementary) calculations.
scientific article; zbMATH DE number 1985409