Generalized Quadrangle

A classification of finite partial linear spaces with a primitive rank 3 automorphism group of almost simple type

A generalized quadrangle $GQ(s,t)$ of order $(s,t)$ is a partial geometry with parameters $(s,t,1)$.

Remark. Note that this definition cannot deduce the definition of generalized polygon, while we can get a equivalent definition of generalized quadrangle from it.

Parameter Conditions

A classification of finite locally 2-transitive generalized quadrangles-Bamberg_Li_Swartz, Lemma 2.7

Let $\mathcal{Q}$ be a finite generalized quadrangle of order $(s,t)$, where $s,t>1$. Then the following hold:

• $|\mathcal{P}|=(s+1)(st+1)$ and $|\mathcal{L}|=(t+1)(st+1)$;
• $s+t$ divides $st(s+1)(t+1)$;
• $t⩽s^2$ and $s⩽t^2$;
• if $s<t^2$, then $s⩽t^2-t$, and if $t<s^2$, then $t⩽s^2-s$.

Remark. See here for more details.

Example: $GQ(2,2)$

The smallest thick GQ is $GQ(2,2)$, and we can get it by taking $q=2$ in the following slide, which comes from here.

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