 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/b/b2/c/solution

A \mapsto (A, 0)

is an injective group homomorphism, so its image is the required copy of

GL_{n} (F)

N = {(I, b) : b \in F^{n}}

, isomorphic to the additive group of

F^{n}

. For any affine map,

(A, b) (I, t) (A, b)^{- 1} = (I, A t) .

Thus conjugation preserves translations, proving

N ⊴ G

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