The $\widetilde P!$-Theorem

The paper under review is part of an ongoing project to give a proof for large parts of the classification of the finite simple groups which is different from the existing one. Moreover, the authors prove some result which is of independent interest.\par They consider the usual amalgam set up, i.e., two groups $M_1,M_2$ which are of characteristic $p$-type, share a common Sylow $p$-subgroup but no nontrivial normal subgroup of $\langle M_1,M_2\rangle$. Now the special assumptions are (i) $Z(M_i)=1$, $i= 1,2$, (ii) There is $Y_{M_i}\triangleleft M_i$ with $M_i/C_{M_i}(Y_i)\cong\text{SL}_3(q_i)$, $\text{Sp}_4(q_i)$, $q_i=p^{n_i}$, or $\text{Sp}_4(2)'$ ($q_i=p=2$) and $[Y_{M_i},O^p(M_i)]$ is the natural module, $i=1,2$. (iii) $C_{M_i}(Y_i)=O_p(M_i)$ or $q_i=2$ and $M_i/O_2(M_i)\cong 3\text{Sp}4(2)$ or $3\text{Sp}_4(2)'$; (iv) There is a 2-dimensional singular subspace $W$ in $[Y_{M_i},O^p(M_i)]$ such that $O^{p'}(N_{M_i}(W))\le M_1\cap M_2$, $i= 1,2$.\par Then the authors show that this setup does just occur in very special situations. Either $p=2$, $O_2(M_i)=Y_{M_i}$ and $M_i/O_2(M_i)\cong\text{Sp}_4(2)'$ or $\text{Sp}_4(2)$, $|Y_{M_i}|=2^4$ or $2^5$ or $q=q_1=q_2$, $p=3$ or $q=5$ and $M_i/O_p(M_i)\cong\text{SL}_3(q)$, and $O_p(M_i)/Y_{M_i}$ and $Y_{M_i}$ are natural $\text{SL}_3(q)$-modules dual to each other.\par This result is similar to the result due to {\it B. Stellmacher} and {\it F. G. Timmesfeld} [Mem. Am. Math. Soc. 649 (1998; Zbl 0911.20024)] but does not follow from that result. This is now used to get a technical result, the $\widetilde P$-theorem. This under certain assumptions, which are technical, says basically the following. Let $G$ be a group with $O_p(G)=1$, $S\in\text{Syl}_p(G)$, of local characteristic $p$, and $\widetilde C$ be a maximal $p$-local containing $N_H(\Omega_1(Z(S)))$. As the generic simple group of local characteristic $p$ is a group of Lie type over a field of characteristic $p$, the aim is to get a geometry for $G$. In this sense then $\widetilde C$ is a maximal parabolic. Now, the authors consider a minimal parabolic $P\nleq\widetilde C$. Under a further technical assumption, they show that there is a unique minimal parabolic $\widetilde P$ containing $S$, which does not normalize $P$ or there is one of the exceptions described by the theorem above. Further, they show that the group generated by $P$ and $\widetilde P$ is a rank 2 Lie group. So if $\widetilde C$ induces a Lie group this result provides us with a building geometry for $\langle P,\widetilde C\rangle$. [Gernot Stroth (Halle)]