On the maximal subgroups of the finite classical groups.

*(English)*Zbl 0537.20023In the article under review the author studies finite groups \(G\), such that \(G_ 0=F^*(G)\) is a classical simple group (i.e. linear, unitary, symplectic or orthogonal over a finite field \(F=GF(q))\). He defines a collection \(\phi_ G\) of subgroups which occur “naturally” in \(G\). Denote by \(\tilde G\) the corresponding (semi-)linear group in \(\Gamma(V)\), where \(V\) is the natural \(F\)-module for \(G_ 0\). Roughly speaking \(\phi_ G\) has members of the following types:

(a) \(N_{\tilde G}(U)\), where \(U\) is a totally singular or a nonsingular subspace of \(V\).

(b) Stabilizers of certain orthogonal decompositions of \(V\).

(c) Stabilizers of decompositions of \(V\) into two totally singular subspaces.

(d) Groups \(N_{\tilde G}(K)\) where \(K\) is a field containing \(F\) with prime index (typical situation: \(SL(m,q^ 2)\subseteq SL(2m,q)\)).

(e) \(F_ 1\subseteq F\) is a subfield of prime index and we have a group of the form \(N_ G(U)\) where \(U\) is an \(n\)-dimensional \(F_ 1\)-space (typical situation: \(SL(m,\root r\of q)\subseteq SL(m,q)\)).

(f) \(N_{\tilde G}(R)\), \(R\) is a group of symplectic type.

(g) \(V\) has the form \(f\) and decomposes as \(V=(V_ 1,f_ 1)\otimes...\otimes(V,f_ t)\) such that \(f\simeq f_ 1\otimes...\otimes f_ t.\) Then we have stabilizers of such decompositions (typical situation: a wreath product of \(\Gamma(V_ 1)\) with \(S^ n\)).

(h) Classical subgroups of dimension \(n\) over \(F\).

The author now proves the following theorem: Assume \(H\) is a maximal subgroup of \(G\) with \(G=HG_ 0\) then \(H\in \phi_ G\) or \(F^*(H)\) is simple and if \(L\) is the covering group of \(F^*(H)\) then \(V\) is an absolutely irreducible \(L\)-module and \(F\) is even the field of definition of the \(L\)-module \(V\).

This important article is a step towards the classification of primitive permutation representations of classical groups. Its objective is to sort out the most natural permutations representations of these groups. For instance this result is an essential tool of a work of Liebeck (to be published) on maximal subgroups \(H\) of classical groups over \(GF(q)\) and dimension \(n\) where \(|H|\geq q^{3n}\).

(a) \(N_{\tilde G}(U)\), where \(U\) is a totally singular or a nonsingular subspace of \(V\).

(b) Stabilizers of certain orthogonal decompositions of \(V\).

(c) Stabilizers of decompositions of \(V\) into two totally singular subspaces.

(d) Groups \(N_{\tilde G}(K)\) where \(K\) is a field containing \(F\) with prime index (typical situation: \(SL(m,q^ 2)\subseteq SL(2m,q)\)).

(e) \(F_ 1\subseteq F\) is a subfield of prime index and we have a group of the form \(N_ G(U)\) where \(U\) is an \(n\)-dimensional \(F_ 1\)-space (typical situation: \(SL(m,\root r\of q)\subseteq SL(m,q)\)).

(f) \(N_{\tilde G}(R)\), \(R\) is a group of symplectic type.

(g) \(V\) has the form \(f\) and decomposes as \(V=(V_ 1,f_ 1)\otimes...\otimes(V,f_ t)\) such that \(f\simeq f_ 1\otimes...\otimes f_ t.\) Then we have stabilizers of such decompositions (typical situation: a wreath product of \(\Gamma(V_ 1)\) with \(S^ n\)).

(h) Classical subgroups of dimension \(n\) over \(F\).

The author now proves the following theorem: Assume \(H\) is a maximal subgroup of \(G\) with \(G=HG_ 0\) then \(H\in \phi_ G\) or \(F^*(H)\) is simple and if \(L\) is the covering group of \(F^*(H)\) then \(V\) is an absolutely irreducible \(L\)-module and \(F\) is even the field of definition of the \(L\)-module \(V\).

This important article is a step towards the classification of primitive permutation representations of classical groups. Its objective is to sort out the most natural permutations representations of these groups. For instance this result is an essential tool of a work of Liebeck (to be published) on maximal subgroups \(H\) of classical groups over \(GF(q)\) and dimension \(n\) where \(|H|\geq q^{3n}\).

Reviewer: U.Dempwolff

##### MSC:

20G40 | Linear algebraic groups over finite fields |

20B15 | Primitive groups |

20D40 | Products of subgroups of abstract finite groups |

20D25 | Special subgroups (Frattini, Fitting, etc.) |

20D06 | Simple groups: alternating groups and groups of Lie type |

##### Keywords:

classical simple groups; orthogonal decompositions; stabilizers; groups of symplectic type; wreath products; covering groups; primitive permutation representations of classical groups; maximal subgroups##### References:

[1] | Aschbacher, M.: On finite groups of Lie type and odd characteristic. J. Alg.66, 400-424 (1980) · Zbl 0445.20008 |

[2] | Aschbacher, M., Scott, L.: Maximal subgroups of finite groups. J. Alg. in press (1984) · Zbl 0549.20011 |

[3] | Aschbacher, M., Seitz, G.: Involutions in Chevalley groups over fields of even order Nagoya Math. J.63, 1-91 (1976) · Zbl 0359.20014 |

[4] | Huppert, B.: Endliche Gruppen Vol. I, Berlin-Heidelberg-New York: Springer 1967 · Zbl 0217.07201 |

[5] | Passman, D.: Permutation, groups. New York: Benjamin 1968 · Zbl 0179.04405 |

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