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 determine 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, which were recently found by the first author (finite case) and by Kedra, Libman and Martin (infinite case).