Camina pairs that are not \(p\)-closed. (Q2339603)

A pair \((G,N\)) is called a Camina pair if \(N\) is a normal subgroup of \(G\) and every element \(g\in G\setminus N\) is conjugate to all of \(gN\). The study of Camina pairs has been directed to those which are not Frobenius groups and it can be divided into three categories: (1) \(G\) is a \(p\)-group, (2) \(G/N\) is a \(p\)-group and \(N\) is not a \(p\)-group, but \(p\) divides \(|N|\), and (3) \(N\) is a \(p\)-group and \(G/N\) is not a \(p\)-group but \(p\) divides \(|G/N|\). Practically all of the known examples of Camina pairs of type (3) are \(p\)-closed, that is, have a normal Sylow \(p\)-subgroup. The only published non-\(p\)-closed examples until now have been for the primes \(2\), \(3\) and \(5\). In this paper, however, the author provides for every prime \(p\) a family of Camina pairs of type (3) where \(G\) is not \(p\)-closed. In particular, since E. B. Kuisch had already proved that for such pairs, \(G\) has \(p\)-length at most 2, then it is obtained that the \(p\)-length equal to 2 can occur for all primes. The construction of the family of examples comes up from suitable group actions on the Sylow \(p\)-subgroups of \(\text{GL}(3,p)\). The proofs involve some character theory such as Gagola characters.