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

Abstract

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\in G $ and $ S $ is a normally generating subset of $ G, $ then we write $ \| g\|_{S} $ for the length of a shortest word in $ \mbox{Conj}_{G}(S^{\pm 1}): = \{h^{-1}sh | h\in G, s\in S \, \mbox{or} \, s{^{-1}}\in S \} $ needed to express $ g. $ For any normally generating subset $ S $ of $ G, $ we write $ \|G\|_{S} = \mbox{sup}\{\|g\|_{S} \, |\, \, g\in G\}. $ Moreover, we write $ \Delta(G) $ for the supremum of all $ \|G\|_{S}, $ 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.