The binary actions of alternating groups
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917564851945472 |
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| author | Gill, Nick Guillot, Pierre |
| author_facet | Gill, Nick Guillot, Pierre |
| contents | Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $Γ(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $Γ(\mathcal{C})$ if $[g,h]=1$ and either $gh^{-1}$ or $hg^{-1}$ is in $\mathcal{C}$.
We prove a lemma that relates the binary actions of the group $G$ to connectivity properties of $Γ(\mathcal{C})$. This lemma allows us to give a complete classification of all binary actions when $G=A_n$, an alternating group on $n$ letters with $n\geq 5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06003 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The binary actions of alternating groups Gill, Nick Guillot, Pierre Group Theory Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $Γ(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $Γ(\mathcal{C})$ if $[g,h]=1$ and either $gh^{-1}$ or $hg^{-1}$ is in $\mathcal{C}$. We prove a lemma that relates the binary actions of the group $G$ to connectivity properties of $Γ(\mathcal{C})$. This lemma allows us to give a complete classification of all binary actions when $G=A_n$, an alternating group on $n$ letters with $n\geq 5$. |
| title | The binary actions of alternating groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2303.06003 |