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The relational complexity of a subgroup 𝐺 of Sym ( Ω ) \mathrm{Sym}({\Omega}) is a measure of the way in which the orbits of 𝐺 on Ω k \Omega^{k} for various 𝑘 determine the original action of 𝐺. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSL n ( F ) \mathrm{PSL}_{n}(\mathbb{F}) and PGL n ( F ) \mathrm{PGL}_{n}(\mathbb{F}) , for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of F n \mathbb{F}^{n} . We also bound the relational complexity of all groups lying between PSL n ( q ) \mathrm{PSL}_{n}(q) and P Γ L n ( q ) \mathrm{P}\Gamma\mathrm{L}_{n}(q) , and generalise these results to the action on 𝑚-spaces for m ≥ 1 m\geq 1 .
The relational complexity of a subgroup 𝐺 of Sym ( Ω ) \mathrm{Sym}({\Omega}) is a measure of the way in which the orbits of 𝐺 on Ω k \Omega^{k} for various 𝑘 determine the original action of 𝐺. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSL n ( F ) \mathrm{PSL}_{n}(\mathbb{F}) and PGL n ( F ) \mathrm{PGL}_{n}(\mathbb{F}) , for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of F n \mathbb{F}^{n} . We also bound the relational complexity of all groups lying between PSL n ( q ) \mathrm{PSL}_{n}(q) and P Γ L n ( q ) \mathrm{P}\Gamma\mathrm{L}_{n}(q) , and generalise these results to the action on 𝑚-spaces for m ≥ 1 m\geq 1 .
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