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 $\Vert g{\Vert }_S$ 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 $\Vert G{\Vert }_S,$ 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).

25 pages