Let V be the 2-dimensional column vector space over a finite field \documentclass{aastex}
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$$\mathbb{F}_q$$
\end{document} (where q is necessarily a power of a prime number) and let ℙq be the projective line over \documentclass{aastex}
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$$\mathbb{F}_q$$
\end{document}. In this paper, it is shown that GL2(q), for q ≠ 3, and SL2(q) acting on V − {0} have the strict EKR property and GL2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GLn(q) acting on \documentclass{aastex}
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$$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$
\end{document} has the EKR property and the derangement graph of PSL2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.
Linear sets in the projective line over the endomorphism ring of a finite field
