On \(p\)-groups having the minimal number of conjugacy classes of maximal size.
There is a problem of I. D. Macdonald of existence of a finite \(p\)-group of odd order with exactly \(p-1\) conjugacy classes of maximal size (e.g., Problem 14.92 in The Kourovka Notebook). The authors produce a 3-group with this property. In fact, they prove two general results (which combine only for \(p=3\)). A conjugacy class of a finite \(p\)-group is called special if it is equal to a coset of the Frattini subgroup; a special class is always of maximal size, and if there is a special class, then all classes of maximal size are special. Theorem~1: For any odd prime \(p\) there is a group of order \(p^{2p+2}\) with exactly \((p-1)^2\) special conjugacy classes. Theorem~2: If there is a finite non-cyclic \(p\)-group with exactly \(2(p-1)\) special conjugacy classes, then there is a finite \(p\)-group with exactly \(p-1\) special conjugacy classes.
scientific article; zbMATH DE number 5610600