By P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi
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1. If G∗ is a simple K-group, x∗ is an involution in G∗ , and C = CG∗ (x∗ ), then the following conditions hold: (i) C ∗ has at most 2 nonsolvable composition factors, unless G∗ is an orthogonal group, in which case C ∗ has at most 4 nonsolvable composition factors; (ii) If G∗ is of Lie type of characteristic r, then each nonsolvable composition factor of C ∗ is of Lie type of characteristic r; and (iii) O(C ∗ ) is cyclic. ] Since the automorphism group of a cyclic group is abelian, the cyclicity of O(C ∗ ) yields the B2 -property as an important corollary.
4, then that deﬁnition requires that the minimal polynomial of g in its action on V be (t − 1)2 , rather than (t − 1)p = tp − 1, as one might expect. Of course if p = 2 and X has even order, any faithful F2 Xmodule is quadratic with respect to any subgroup of X of order 2. Thus the notion is nontrivial only for odd p or |A| ≥ 4. In these cases, quadratic modules are rather rare among all isomorphism types. For example, if p ≥ 5 and G is solvable with Op (G) = 1, then G has no quadratic modules.
The Odd Order Theorem showed that every (nonabelian) simple group must have even order and hence necessarily must contain involutions. The N -group theorem demonstrated the power and the potential of local group-theoretic analysis for treating broad classiﬁcation theorems. The “2-rank ≤ 2” theorem disposed once and for all of the “smallest” simple groups. The inductive family of groups of “sectional 2-rank ≤ 4” had been introduced [MacW1] as a means of treating the noninductive problem of determing the simple groups with a nonconnected Sylow 2-subgroup (a group S is said to be connected if any two four-subgroups17 of S can be included in a chain of four-subgroups of S in which every two successive 17 A four-group is a group isomorphic to Z2 × Z2 .