Finite Classical Groups [Lecture notes] by Nick Gill

By Nick Gill

(London Taught direction heart 2013)

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Finite Classical Groups [Lecture notes]

(London Taught direction middle 2013)

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A , b , . . , b ∈ k, a     0 � 1 2r−1 1 2r−2 b   � 2r−2  �    −ai , if i ≤ r − 1;   .. � . (17) G�w1 � = g :=  . � bi = A .  otherwise; ai , �          0 b1   a ∈ k ∗ , A ∈ Sp2r−2 (k)     0 0 ··· 0 a−1 Now there is a natural epimomorphism G�w1 � → Sp2r−2 (k) × GL1 (k), g �→ (A, a) e: Q2 and the kernel of this map is the group   1 a1 · · · a2r−2 a2r−1      0 b2r−2     .. (18) Q := g :=  . I .      0 b1    0 0 ··· 0 1  � . , a2r−1 , b1 , .

Automorphisms. The true significance of Tits’ classification of the finite spherical buildings lies in their automorphism groups. We saw earlier that Aut(PGn−1 (q)) = PΓLn (q) which, provided n ≥ 2 or q ≥ 4, is an almost simple group with simple normal subgroup PSLn (q). In the next few sections we will see that the automorphism groups of the classical polar spaces are (generally speaking) almost simple groups with simple normal subgroup equal to a classical group. The beauty of Tits’ classification is that the automorphism groups of the spherical buildings are (generally speaking) almost simple groups with simple normal subgroup equal to a finite group of Lie type.

Isom(Q) = �rv | Q(v) �= 0�, provided Isom(Q) �= O+ 4 (2). Now our definition is as follows: • Suppose that q is even and that Isom(Q) �= O+ 4 (2). We can assume that n is even by Lemma 68 and thus, by (E109), Oεn (q) = SOεn (q) and by Lemma 72, every element of SOεn (q) can be written as a product of reflections. Now the subgroup of S consisting of products of an even number of reflections has index 2 in SOεn (q) and this is the group Ωεn (q). It is not a priori clear that this action yields an index 2 subgroup - the next exercise shows that it is true when ε = +.

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