 Solution

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

Every pair factors as

(A, b) = (I, b) (A, 0)

, and a linear map which is also a translation fixes zero and has linear part

I

, hence is the identity. Thus

G = N H

and

N \cap H = {1}

. The conjugation formula in (c) says that the action needed for reconstruction is simply the usual matrix action

A : t \mapsto A t

on the additive group

F^{n}

. Therefore

G ≅ F^{n} ⋊ GL_{n} (F) .

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