The conjugacy diameters of non-abelian finite p-groups with cyclic maximal subgroups

Let G be a group. A subset S of G is said to normally generate G if G is the normal closure of S in G. In this case, any element of G can be written as a product of conjugates of elements of S and their inverses. If g is an element of G and S is a normally generating subset of G, then we write kgkS for the length of a shortest word in ConjG(S +/- 1) := {h-1sh|h is an element of G, s is an element of S ors-1 is an element of S } needed to express g. For any normally generating subset S of G, we write kGkS = sup{kgkS | g is an element of G}. Moreover, we write Delta(G) for the supremum of all kGkS, where S is a finite normally generating subset of G, and we call Delta(G) the conjugacy diameter of G. In this paper, we derive the conjugacy diameters of the semidihedral 2 -groups, the generalized quaternion groups and the modular p -groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups.