Let $\Gamma (G;(G\_i)\_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to find all the elements $\rho\_i$ in the minimal parabolics subgroups $G\_{I-\{i\}}$ such that $[ \langle H ,\rho\_i \rangle : H] = 2$.


I have been given an idea: consider the quotient $G\_{I-\{i\}}/B$ together with the canonical homomorphism $\phi\_i : G\_{I-\{i\}} \to G\_{I-\{i\}}/B$. Then keep all the elements $\rho\_i \in G\_{I-\{i\}}-B$ such that $\phi\_i(\rho\_i)$ is an involution. In order to find them, take all involutions in $G\_{I-\{i\}}/B$, then take their preimage by $\phi\_i$ and it then suffices to multiply the preimage by the elements in the kernel of $\phi\_i$ and keep those which normalize $H$ to get the elements satisfying the condition given above.


However, I have two problems. First, I don't really see why this works. Second is that I have no guarantee that $B$ is normal in each minimal parabolic subgroup. Worse, most of the time it is not. Hence, the set of right cosets of $B$ in $G\_{I-\{i\}}$ does not have a group structure. Therefore, I have some problem with this solution.


Does anyone have a suggestion?


Thanks in advance.



Find elements $\rho_i$ such that $H < B : [\langle H, \rho_i \rangle : H ] = 2$
First compute the normalizer $N\_{P\_i}(H)$ of $H$ in each minimal parabolic subgroup $P\_i$ of $\Gamma$. Then consider the quotient $N\_{P\_i}(H)/H$ and the natural homomorphism $\varphi\_i:N\_{P\_i}(H) \to N\_{P\_i}(H)/H$. Now define the set 
$$S\_i := \{ \rho \in N\_{P\_i}(H) \mid (\varphi\_i(\rho))^2 = 1 \} $$
 of elements of $N\_{P\_i}(H)$ each of which has an image by $\varphi\_i$ in $N\_{P\_i}(H)/H$ that is an involution. We then have $[\langle H,\rho \rangle : H] = 2$ for any $\rho \in S\_i$.


Indeed, since $(\varphi\_i(\rho))^2=1$, one has $H\rho^2=H$ and since $\rho \in N\_{P\_i}(H)$, one has $\rho H = H \rho$. Now take $x \in \langle H,\rho \rangle$. One can write $x=h\_1 \rho^{p\_1} \dots h\_k \rho^{p\_k}$ for some $h\_i \in H$ and some powers $p\_i \in \mathbb{Z}$, where $i \in \{1,\ldots, k\}$. Let us consider $Hx= Hh\_1 \rho^{p\_1} \dots h\_k \rho^{p\_k}$. Using the fact that $\rho$ normalizes $H$, one gets $Hh\_1 \rho^{p\_1} \dots h\_k \rho^{p\_k} = H \rho^{p\_1} \dots \rho^{p\_k}$. One concludes quickly that $Hx=H$ or $Hx=H\rho$, i.e. there are only two cosets of $H$ in $\langle H,\rho \rangle$.


