h4.pdf - Math 155 Designs and groups Handout#4 Simplicity of PSL2(F(|F | > 4 and PSLn(F(n > 3 — Outline 0 Let F be a finite field of q elements PSLn(F

h4.pdf - Math 155 Designs and groups Handout#4 Simplicity...

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Math 155: Designs and groups Handout #4: Simplicity of PSL 2 ( F ) ( | F | 4) and PSL n ( F ) ( n 3) — Outline 0. Let F be a finite field of q elements. PSL n ( F ) is a normal subgroup [indeed the commutator subgroup, but we won’t need this] of PGL n ( F ) with index gcd( n, q - 1), and is generated by “transvections” because SL n ( F ) is; indeed even coordinate transvections suffice. (A coordinate transvection is a matrix with 1’s on the diagonal and a single nonzero off-diagonal entry. A linear transformation T : F n F n that is of that form for some choice of basis is a transvection; an equivalent coordinate-free criterion is: T - I has rank 1 and square zero.) When n = 2 the transvections in PSL 2 ( F ) are precisely the fractional linear transformations of P 1 ( F ) with exactly one fixed point; if that point is , the transformation is x x + c for some c F * . 1. Let G = PSL 2 ( F ) and assume H is a normal subgroup of G . If H contains a transvection then it contains all of them, and thus coincides with G . (The G -conjugates of x x + c
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