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I want to prove that $GL(2n, q)$ where $q$ is even, has an abelian subgroup of order $q^{2n - 1}$. please help me.

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The matrices consisting of 1's on the main diagonal and 0's everywhere else except in the last column form an abelian subgroup of order $q^{2n-1}$. In fact, if the last column (other than the 1 at the bottom) is treated as a vector, the multiplication operation is just addition of vectors.

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