The Subgroup Structure of the Finite Classical Groups
Cambridge University Press, Apr 26, 1990 - Mathematics - 303 pages
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.
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Motivation and Setting for the Results
Basic Properties of the Classical Groups
The Statement of the Main Theorem
The Structure and Conjugacy of the Members of C
Properties of the Finite Simple Groups
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2-space absolutely irreducible algebraic appears in Table Aschbacher’s assume that H automorphism bilinear form centralizes Chapter classical groups completes the proof Consequently contradiction Corollary covering group deduce deﬁne deﬁnition deleted permutation module denotes described dim(V dim(W dimension embed embedding ﬁeld ﬁnd ﬁnite ﬁx ﬁxes follows from Lemma follows from Proposition GL(V groups of Lie groups of type H5 is non-local hence homomorphism implies induces irreducibly isometric isomorphic Lemma Levi factor Lie rank Lie type Lie(p maximal subgroups minimal Moreover non-abelian non-singular non-singular vector non-square non-trivial non-zero notation obtain orthogonal groups orthonormal orthonormal basis overgroups parabolic subgroup preimage proof of Proposition Proposition 5.3.7 prove q is odd q odd quasisimple quasisimple groups reﬂection representation result satisﬁes sgn(Q shows simple groups space spin module spinor norm Spm(q sporadic groups stabilizer standard basis structure Suppose ﬁrst tensor decomposition tensor product unique write