Frobenius partitions in extraspecial groups (Q1282313)

Let \(G\) be a group with a partition \(\mathbf F\) into a distinguished normal subgroup \(N\) and a family \((H_i)\) of further subgroups such that \(H_iN=G\) for all \(i\in I\). The authors call \(\mathbf F\) a Frobenius partition if the \(H_i\) form an orbit under a subgroup of \(\Aut G\) (in its natural action). They use commutative division algebras to construct examples; in particular, in the finite case they obtain examples where \(N\) has order \(q^n\) and all the \(H_i\) have order \(q^{n+1}\); here \(q\) is an arbitrary odd prime power, \(n\) a natural number and \(G\) a generalized extraspecial group.