Letting $\mathbf{k}$ be a field with $q$ elements. How many short exact sequences of vector spaces are there $$\mathbf{k} \hookrightarrow \mathbf{k}^2 \twoheadrightarrow \mathbf{k}\,?$$ My brain is telling me something different than my GAP program, and I'm trying to figure out which one is lying to me. ;)
2026-03-31 05:40:08.1774935608
How many short exact sequences are there $\mathbf{k} \hookrightarrow \mathbf{k}^2 \twoheadrightarrow \mathbf{k}$?
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There are $q^2 - 1$ choices for the injection. Once the injection is chosen, pick basis $a, b$ for $k^2$ where $a$ is in the image of the injection and $b$ isn't. Then the surjection must send $a$ to $0$ (since the sequence is exact) and $b$ to a non-zero element. This gives us a total of $(q^2 - 1)(q - 1)$ choices.