An \(F_1\)-module for the \(3U_4(3)\)-amalgam. (English) Zbl 0918.51016
An important part of the author’s PhD thesis [Some semiclassical parabolic systems of rank 4, Dissertation, Martin-Luther-Universität Halle-Wittenberg (1997)] is the determination of \(F_1\)-moduls for groups of type \(3 U_4(3)\). The result is as follows. Assume that the set of groups \(\{P_1, P_2, P_3\}\) forms a semi-classical parabolic system for the group \(G = \langle P _1,P _2,P _3 \rangle \neq G_i = \langle P _j,P _k \rangle\) for \(\{i,j,k\} = \{1,2,3\}\), s.t. \(S = P _1 \cap P _2 \cap P _3 \in Syl _2(G_i)\), \(P _i/O _2(P _i) \cong S_3\) (\(i = 1,2,3\)), \(G_i /O _2(G _i) \cong A_6, S_6, 3 A_6\) or \(3 S_6\) (\(i = 1,3\)) and \(G_2 /O _2(G _2) \cong S_3 \times S_3\). Furthermore, assume that \(2^4 \leq \mid O_2(G_i) \mid \leq 2^5\) and \(C_{O_2(G_i)}(G _i) = 1\) for \(i = 1,3\) and \(\Omega_1(Z(S))\) is normal in \(G_2\).
Finally let \(V\) be an irreducible nontrivial \(GF(2)\)-module for \(G\). Then it is shown that \(V\) is not an \(F\)-module and if \(V\) is an \(F_1\)-module, then \(V\) is 12-dimensional and \(G/C_G (V) \cong 3 U_4(3)2\) or \(3 U_4(3)4\).
Finally let \(V\) be an irreducible nontrivial \(GF(2)\)-module for \(G\). Then it is shown that \(V\) is not an \(F\)-module and if \(V\) is an \(F_1\)-module, then \(V\) is 12-dimensional and \(G/C_G (V) \cong 3 U_4(3)2\) or \(3 U_4(3)4\).
Reviewer: S.Heiss (Halle)
MSC:
| 51E24 | Buildings and the geometry of diagrams |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20F65 | Geometric group theory |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
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