 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2008/a/2/b/iii/solution

Relative to the three successive blocks of sizes

1, 2, 2

, the stabilizer consists of the block upper triangular matrices

a 00 B D 0 C E F, a \in K^{\times}, D, F \in GL (2, K) .

The off-diagonal blocks contain

2 + 2 + 4 = 8

arbitrary field entries. Thus its order is

(q - 1) ∣ GL (2, q) ∣^{2} q^{8}

∣Ω∣ = \frac{\prod _{j = 0}^{4} ( q ^{5} - q ^{j} )}{q ^{8} ( q - 1 ) ( q ^{2} - 1 ) ^{2} ( q ^{2} - q ) ^{2}} .

Equivalently, choose the line first and then a two-dimensional subspace of the four-dimensional quotient:

∣Ω∣ = \frac{q ^{5} - 1}{q - 1} \frac{( q ^{4} - 1 ) ( q ^{4} - q )}{( q ^{2} - 1 ) ( q ^{2} - q )} .

This also checks the stabilizer count independently.

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