Let $q$ be a power of a prime number and $V$ be a $2$-dimensional column vector space over a finite field ${\mathbb{F}}_{q}$. 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 ${\mathcal{G}}_{\langle w\rangle}$, where ${\mathcal{G}}_{\langle w\rangle}=\{M\in G\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\forall v\in V,Mv-v\in \langle w\rangle \}$, 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.

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@article{CRMATH_2022__360_G5_497_0, author = {Ahanjideh, Milad}, title = {On the {Largest} intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups}, journal = {Comptes Rendus. Math\'ematique}, pages = {497--502}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, number = {G5}, year = {2022}, doi = {10.5802/crmath.320}, language = {en}, url = {http://www.numdam.org/articles/10.5802/crmath.320/} }

TY - JOUR AU - Ahanjideh, Milad TI - On the Largest intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups JO - Comptes Rendus. Mathématique PY - 2022 SP - 497 EP - 502 VL - 360 IS - G5 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.320/ DO - 10.5802/crmath.320 LA - en ID - CRMATH_2022__360_G5_497_0 ER -

%0 Journal Article %A Ahanjideh, Milad %T On the Largest intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups %J Comptes Rendus. Mathématique %D 2022 %P 497-502 %V 360 %N G5 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.320/ %R 10.5802/crmath.320 %G en %F CRMATH_2022__360_G5_497_0

Ahanjideh, Milad. On the Largest intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups. Comptes Rendus. Mathématique, Volume 360 (2022) no. G5, pp. 497-502. doi : 10.5802/crmath.320. http://www.numdam.org/articles/10.5802/crmath.320/

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