On the Largest intersecting set in ${\mathrm{GL}}_2\left(q\right)$ and some of its subgroups

Ahanjideh, Milad

Comptes Rendus. Mathématique, Tome 360 (2022) no. G5, pp. 497-502.

Let $q$ be a power of a prime number and $V$ be a $2$-dimensional column vector space over a finite field ${𝔽}_q$. Assume that ${\mathrm{SL}}_2\left(V\right)<G\le {\mathrm{GL}}_2\left(V\right)$. In this paper we prove an Erdős–Ko–Rado theorem for intersecting sets of G and we show that every maximum intersecting set of $G$ is either a coset of the stabilizer of a point or a coset of ${𝒢}_{\langle w\rangle }$, where ${𝒢}_{\langle w\rangle }=\{M\in G:\forall v\in V,Mv-v\in \langle w\rangle \}$, for some $w\in V\setminus \left\{0\right\}$. It is also shown that every intersecting set of $G$ is contained in a maximum intersecting set.

Classification : 05D05, 20B35

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