The conjugacy diameters of non-abelian finite $p$-groups with cyclic maximal subgroups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916096014024704 |
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| author | Aseeri, Fawaz Kaspczyk, Julian |
| author_facet | Aseeri, Fawaz Kaspczyk, Julian |
| contents | 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 $Δ(G)$ for the supremum of all $\|G\|_{S},$ where $S$ is a finite normally generating subset of $G,$ and we call $Δ(G)$ 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09645 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The conjugacy diameters of non-abelian finite $p$-groups with cyclic maximal subgroups Aseeri, Fawaz Kaspczyk, Julian Group Theory Combinatorics 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 $Δ(G)$ for the supremum of all $\|G\|_{S},$ where $S$ is a finite normally generating subset of $G,$ and we call $Δ(G)$ 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). |
| title | The conjugacy diameters of non-abelian finite $p$-groups with cyclic maximal subgroups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2401.09645 |