Modules for Goldschmidt amalgams. (Moduln für Goldschmidt-Amalgame.) (German) Zbl 0805.20020
Berlin: FU Berlin, FB 19/Math. 110 S. (1991).
Let \({\mathcal A} = (P_ 1,P_ 2,B)\), \(B \leq P_ 1 \cap P_ 2\) be a finite primitive amalgam of index \((3,3)\). D. M. Goldschmidt [Ann. Math., II. Ser. 111, 377-406 (1980; Zbl 0475.05043)] classified all such amalgams. There are exactly fifteen pairwise nonisomorphic finite primitive amalgams of index (3,3), six of which are irreducible. The irreducible amalgams are weak BN-pairs (as \(P_ 1\) and \(P_ 2\) are 2- constrained iff \(\mathcal A\) is irreducible).
In a paper of A. Chermak [Commun. Algebra 14, 591-665 (1986; Zbl 0593.20027)] all faithful completions \(G\) of finite primitive amalgams \({\mathcal A} = (P_ 1,P_ 2,B)\) of index (3,3) possessing a faithful \(F\)- module \(V\) are determined. Moreover, for every such completion \(G\) the \(F\)-modules for \(G\) are described.
In the work under review, the author gives a similar classification of all pairs \((G,V)\), \(G\) as above acting faithfully on a strong \(SC+1\)- module \(V\), in the case that \(\mathcal A\) is irreducible. (\(V\) is called a strong \(SC+1\)-module for \(\mathcal A\) if there exists \(A \leq B\), \(A \neq 1 = \Phi(A)\) with \(| V :C_ V(U)| \leq | U| | A|\) for all \(U \leq A\).) A knowledge of strong \(SC+1\)-modules is of interest in the study of minimal parabolic systems, see for example [G. Stroth, S. K. Wong, Commun. Algebra 15, 751-790 (1987; Zbl 0628.20018)].
The bulk of the paper consists of a detailed case by case analysis of the possible amalgams and modules. The classification contains too many cases to be reproduced here, but we may quote the following corollary: Let \((G,V)\) be as above and assume \(B \in \text{Syl}_ 2(G)\). Then \(G\) is isomorphic to \(L_ 3(2)\), \(A_ 6\), \(3A_ 6\), \(S_ 6\), \(3S_ 6\), \(G_ 2(2)'\) or \(G_ 2(2)\) and all composition factors of \(V\) are trivial and natural modules for \(G\). The last section contains some results of independent interest. The author proves the uniqueness of completions \(G\) of primitive amalgams of type \(G_ 2(2)'\) of \(G_ 2(2)\) (\(M_{12}\) or \(\text{Aut}(M_{12})\)) acting faithfully on a \(GF(2)G\)- module \(V\) of dimension \(\leq 6\) (of dimension \(\leq 10\), respectively). Finally he shows that \(SL_ 3(2^ n)\) is a completion of the primitive amalgam of type \(L_ 3(2)\) for \(n \neq 2\). This last result also appeared in [Arch. Math. 60, No. 5, 478-481 (1993; Zbl 0789.20024)].
In a paper of A. Chermak [Commun. Algebra 14, 591-665 (1986; Zbl 0593.20027)] all faithful completions \(G\) of finite primitive amalgams \({\mathcal A} = (P_ 1,P_ 2,B)\) of index (3,3) possessing a faithful \(F\)- module \(V\) are determined. Moreover, for every such completion \(G\) the \(F\)-modules for \(G\) are described.
In the work under review, the author gives a similar classification of all pairs \((G,V)\), \(G\) as above acting faithfully on a strong \(SC+1\)- module \(V\), in the case that \(\mathcal A\) is irreducible. (\(V\) is called a strong \(SC+1\)-module for \(\mathcal A\) if there exists \(A \leq B\), \(A \neq 1 = \Phi(A)\) with \(| V :C_ V(U)| \leq | U| | A|\) for all \(U \leq A\).) A knowledge of strong \(SC+1\)-modules is of interest in the study of minimal parabolic systems, see for example [G. Stroth, S. K. Wong, Commun. Algebra 15, 751-790 (1987; Zbl 0628.20018)].
The bulk of the paper consists of a detailed case by case analysis of the possible amalgams and modules. The classification contains too many cases to be reproduced here, but we may quote the following corollary: Let \((G,V)\) be as above and assume \(B \in \text{Syl}_ 2(G)\). Then \(G\) is isomorphic to \(L_ 3(2)\), \(A_ 6\), \(3A_ 6\), \(S_ 6\), \(3S_ 6\), \(G_ 2(2)'\) or \(G_ 2(2)\) and all composition factors of \(V\) are trivial and natural modules for \(G\). The last section contains some results of independent interest. The author proves the uniqueness of completions \(G\) of primitive amalgams of type \(G_ 2(2)'\) of \(G_ 2(2)\) (\(M_{12}\) or \(\text{Aut}(M_{12})\)) acting faithfully on a \(GF(2)G\)- module \(V\) of dimension \(\leq 6\) (of dimension \(\leq 10\), respectively). Finally he shows that \(SL_ 3(2^ n)\) is a completion of the primitive amalgam of type \(L_ 3(2)\) for \(n \neq 2\). This last result also appeared in [Arch. Math. 60, No. 5, 478-481 (1993; Zbl 0789.20024)].
Reviewer: S.Heiss (Halle)
MSC:
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 20C20 | Modular representations and characters |
| 20E42 | Groups with a \(BN\)-pair; buildings |
| 20C33 | Representations of finite groups of Lie type |
| 20D05 | Finite simple groups and their classification |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20F65 | Geometric group theory |
| 20G40 | Linear algebraic groups over finite fields |
| 20F05 | Generators, relations, and presentations of groups |
