Articles publiés
The binary actions of alternating groups
Nick Gill1 ; Pierre Guillot2
1 School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK
2 IRMA, 7 rue René Descartes, 67084 Strasbourg, France
Confluentes Mathematici, Tome 17 (2025), pp. 73-89

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $\Gamma (\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $\Gamma (\mathcal{C})$ 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 $\Gamma (\mathcal{C})$. This lemma allows us to give a complete classification of all binary actions when $G=A_n$, an alternating group on $n$ letters with $n\ge 5$.

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DOI : 10.5802/cml.101
Classification : 20D06, 20B25, 20B10
Keywords: permutation group, binary action, relational complexity, alternating group

Nick Gill 1 ; Pierre Guillot 2

1 School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK
2 IRMA, 7 rue René Descartes, 67084 Strasbourg, France
Licence : CC-BY-NC-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Nick Gill; Pierre Guillot. The binary actions of alternating groups. Confluentes Mathematici, Tome 17 (2025), pp. 73-89. doi: 10.5802/cml.101

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